From 68342deffd35316b5fe4544a8b595c5c6dfe9c83 Mon Sep 17 00:00:00 2001 From: cb341 Date: Sun, 6 Sep 2026 13:39:26 +0200 Subject: [PATCH 01/14] Initial draft, pixelated images --- _layouts/base.html | 5 + _posts/2026-09-06-shape-of-logic.md | 137 ++++++++++++++++++ assets/blog/shape_of_logic_division_8x8.png | Bin 0 -> 3384 bytes assets/blog/shape_of_logic_lshft_8x8.png | Bin 0 -> 3262 bytes assets/blog/shape_of_logic_multiply_8x8.png | Bin 0 -> 96451 bytes assets/blog/shape_of_logic_tc_major.png | Bin 0 -> 554111 bytes .../blog/shape_of_logic_tc_major_nand_8x8.png | Bin 0 -> 10929 bytes assets/blog/shape_of_logic_tooltip.png | Bin 0 -> 60904 bytes 8 files changed, 142 insertions(+) create mode 100644 _posts/2026-09-06-shape-of-logic.md create mode 100644 assets/blog/shape_of_logic_division_8x8.png create mode 100644 assets/blog/shape_of_logic_lshft_8x8.png create mode 100644 assets/blog/shape_of_logic_multiply_8x8.png create mode 100644 assets/blog/shape_of_logic_tc_major.png create mode 100644 assets/blog/shape_of_logic_tc_major_nand_8x8.png create mode 100644 assets/blog/shape_of_logic_tooltip.png diff --git a/_layouts/base.html b/_layouts/base.html index fbd8346..de3f19a 100644 --- a/_layouts/base.html +++ b/_layouts/base.html @@ -98,6 +98,11 @@ } p img { width: 100%; height: auto; } + img.pixelated-image { + width: 100%; + height: auto; + image-rendering: pixelated; + } p img.about-avatar { float: right; width: min(145px, 32vw); diff --git a/_posts/2026-09-06-shape-of-logic.md b/_posts/2026-09-06-shape-of-logic.md new file mode 100644 index 0000000..c34eeab --- /dev/null +++ b/_posts/2026-09-06-shape-of-logic.md @@ -0,0 +1,137 @@ +--- +title: "Shape of Logic" +description: "" +tags: [] +math: true +--- + + +8 by 8 division visualization + +8 by 8 left-shift visualization + +8 by 8 multiplication visualization + +![foo](/assets/blog/shape_of_logic_tc_major.png) + +8 by 8 NAND visualization + +![foo](/assets/blog/shape_of_logic_tooltip.png) + +context: + +playing around the major rework of the campagin of turing complete 2, +shape_of_logic_tc_major.png + +i workded on the next version of the ALU. + +the new ALU includes next to the usual +- one cycle addition, subtraction +- multiplication + + +also division. +for which a new circuit is needed. + +i didn't know how to build the circuit so i went searching for pattterns in bits. + +i started with a simple script that yields an ascii truth table. +Given two inputs A and B as binary strings, the resulting A / B (integer division, mod) + +... + +I didn't quite see the pattern yet in 1 and 0s + +So I tried different unicode values for 1 and zero, leaving zero empty, focusing only on zero... + +then i treid generating images +first black and white. +1 = white +0 = black + +this was neat but not quite there yet. + +i thought about other ways to color. +what about one pixel per BYTE instead of BIT + +there we already get more interesting patterns + +but how to color? +naive is graycscale 0 = black, 255 = white + +this already helps. + +but we are still not painting the full pictue +the alu is 16 bit. + +we need more resolution. + +i tried generating images for full 8x8 bit combinations but this was wayyyb too slow. + + +so i turned to an LLM to see if codex could generate a pattern viewer. +we started at image rendering clietn side +we eventually moved to shaders that evaluate logic at level of individual pixels. +viewport sizing, toolitps with color encoded bytes. + +8 bits is nice but can we go larger? + +8 bits +2^8, 256x256 = 65'536 pixels, 8 bits per pixel +524'288 bits of raw, not tiny but managable. + +2^16 = 65'536x65'536 = 4'294'967'296 pixels +with 8 bits per pixel yields 34'359'738'368 which is simply not managable. +So i thought about ways we could achieve visualizations without reaching over memeory limits. then it hit me. why don't we simply provide a viewport to vierw the image one part at a time? + +The interactive viewport led to shaders. +you cannot download the full image, but you can see an approximation. +it shows more detail than the 8x8 one but shows patterns only marginally better. + + +i thought about hey why are we even looking at patterns of division now? +what would NAND gates look like? What about addition? do we see the carry pattern, the overflow. Can we see the chaotic nature of multiplication? + +Palletes +- MSB invert (let leading bit determine sign of number, negative numbers are shown inverted ) +- nibble rgb (each nibble = 4bits gets own part of the specturm, patterns are reconglizable not solely on value but in pattern within the number. ) +- vibrant (tested some colors that worked, colors that didn't work) +- monochrome - simple, stupid + +What if we play with color pallettes + + +Now onto division + +{division} + +What you can clearly see is the diagonal stripe +we round results, discard reminder. +the tolerance for identity grows with size + +most change hapesn with smal vlaues, +regulartiy with odd/even + +bands + + +we can see upper right triangle of the matrix is empty. +if we divide x by y with y being larger, we get zero. +this checks out. + + +NOTES/IDEAS + +- Include MY logic gate cirucits for NAND,NOR,MUL,ADD,SUB + +Didn't quite find the pattern yet but I will once i get back to spending time in the emulator + 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shape_of_logic_div_8x8.png} | Bin ...ply_8x8.png => shape_of_logic_mul_8x8.png} | Bin assets/blog/shape_of_logic_toolbar.png | Bin 0 -> 107113 bytes 5 files changed, 192 insertions(+), 78 deletions(-) create mode 100644 assets/blog/shape_of_logic_add_8x8.png rename assets/blog/{shape_of_logic_division_8x8.png => shape_of_logic_div_8x8.png} (100%) rename assets/blog/{shape_of_logic_multiply_8x8.png => shape_of_logic_mul_8x8.png} (100%) create mode 100644 assets/blog/shape_of_logic_toolbar.png diff --git a/_posts/2026-09-06-shape-of-logic.md b/_posts/2026-09-06-shape-of-logic.md index c34eeab..b0fe20f 100644 --- a/_posts/2026-09-06-shape-of-logic.md +++ b/_posts/2026-09-06-shape-of-logic.md @@ -1,137 +1,251 @@ --- title: "Shape of Logic" -description: "" -tags: [] +description: "Every byte pair of an 8-bit operator fits in one 256x256 image. The pictures show tiling, carry, and overflow as shapes." +tags: ["logic", "visualization", "hardware", "first principles"] math: true --- +8 by 8 multiplication visualization -8 by 8 division visualization +*multiplication, every pair of bytes. [^mul-link]* -8 by 8 left-shift visualization +what is this? + +## The idea + +an 8-bit operator takes two bytes and returns one. that is $256 \times 256 = 65{,}536$ input pairs, and the whole thing fits on a screen. + +put $A$ on the x axis, $B$ on the y axis, and colour the pixel at $(A, B)$ by the result of $A \mathbin{\mathrm{op}} B$. the origin sits bottom-left, so $A$ grows rightwards and $B$ upwards. + +that image is the operator's truth table over that domain, every input pair at once. + +the colour is per-bit rather than a value ramp: each of the 8 result bits owns a hue, and a pixel mixes the hues of the bits that are set. that matters later, because two numerically close values can look nothing alike. + +## Starting in ascii: 4-bit division -8 by 8 multiplication visualization +the first version printed characters. this is division, $A / B$, at 4 bits, with set bits as `@` and clear bits blank. -![foo](/assets/blog/shape_of_logic_tc_major.png) +``` + A\B 0000 0001 0010 0011 0100 0101 0110 0111 1000 1001 1010 1011 1100 1101 1110 1111 + 0000 + 0001 @ + 0010 @ @ + 0011 @@ @ @ + 0100 @ @ @ @ + 0101 @ @ @ @ @ @ + 0110 @@ @@ @ @ @ @ + 0111 @@@ @@ @ @ @ @ @ + 1000 @ @ @ @ @ @ @ @ + 1001 @ @ @ @@ @ @ @ @ @ @ + 1010 @ @ @ @ @@ @ @ @ @ @ @ @ + 1011 @ @@ @ @ @@ @ @ @ @ @ @ @ @ + 1100 @@ @@ @ @@ @ @ @ @ @ @ @ @ + 1101 @@ @ @@ @ @@ @ @ @ @ @ @ @ @ @ + 1110 @@@ @@@ @ @@ @ @ @ @ @ @ @ @ @ @ + 1111 @@@@ @@@ @ @ @@ @@ @ @ @ @ @ @ @ @ @ @ +``` + +leaving the zeros blank is what makes the shape appear. the empty upper-right region is already there, and so is the staircase running down the diagonal. + +at 4 bits this is readable. at 8 bits it is 256 rows of 256 columns and the terminal stops being the right tool. + +## Bitwise operators tile 8 by 8 NAND visualization -![foo](/assets/blog/shape_of_logic_tooltip.png) +*NAND, monochrome. [^nand-link]* + +the picture is made of nested copies of itself. a quadrant repeats the whole, and inside it the same block repeats again, down to single pixels. + +the reason is that bitwise operators have no communication between bit positions. result bit $k$ depends on $A_k$ and $B_k$ and nothing else. so the 8-bit table is four copies of the 7-bit table, which is four copies of the 6-bit table, all the way down. + +the self-similarity goes as deep as the bit count allows. for all 65,536 pairs, the low nibble of $A \mathbin{\overline{\wedge}} B$ depends only on the low nibbles of $A$ and $B$. + +## Addition breaks the tiling + +8 by 8 addition visualization + +*addition, nibble RGB. the blocks are gone. [^add-link]* + +carry. bit $k$ of $A + B$ depends on every bit below $k$. the columns are coupled, the independence is gone, and the nested blocks go with it. the same nibble test that holds for NAND fails for addition, exactly as the carry chain predicts. + +what replaces the blocks is diagonal bands. $A + B$ is constant along an anti-diagonal, so each band is one result value. + +this is the seam where logic stops and arithmetic begins, and it is visible. + +the palette here splits the result by nibble, so the high nibble drives blue and the low nibble drives red. the leading bit lives in that high nibble, which makes the cool bands the results a signed reading calls negative, and the warm ones positive. + +read along a diagonal and the bands alternate. crossing one edge flips the sign: two positive numbers add to a negative one, two negatives add to a positive. that is signed overflow, and it is not rare. it happens in exactly 25% of all pairs, 8,128 cells where positive plus positive goes negative and 8,256 where negative plus negative goes positive. + +the fine striping is because this is the 16-bit view. the bands survive the widening, which is the first hint that these are properties of the operator rather than of the byte. + +## Overflow has a shape + +there are 256 bands, and each one is a residue class $[r]_{256}$. counting cells, every class holds exactly 256 of the 65,536 pixels, uniformly. + +the bands are the cosets of $256\mathbb{Z}$ in $\mathbb{Z}$, drawn to scale. + +take $148 + 138$. the true sum is $286$. the byte says $30$. + +$$286 \equiv 30 \pmod{256}$$ + +that wrap is a band edge. every place the diagonal stripe restarts, an overflow just happened, and there are 255 such edges at even spacing across the image. + +overflow is usually taught as a bug class, a thing that bites you at the boundary. the picture gives it a shape: $\mathbb{Z}/256\mathbb{Z}$, the quotient, made visible. the boundary is not an edge case sitting at the end of the range. it is a wall you hit once per band, over and over. + +## The seam that isn't there + +if two's complement were a property of the data, there would be a visible seam at 128 where numbers go negative. + +there is none. across all 65,536 pairs the result bits of signed and unsigned addition are identical everywhere. the same holds for subtraction and for multiplication. + +this is the whole trick of two's complement. it is a reading convention applied to bits. the hardware adder does not know which convention you are using, and that is exactly why the representation won. + +the image cannot show a distinction that is not in the data. + +you can make a seam appear by choosing the MSB-invert palette, which flips the colour when the leading bit is set. that seam lives in the palette. + +## Owning the rendering + +a colour map can invent structure that the data does not have, so this deserves an answer. + +the palette is a per-bit layering, and switching it changes which patterns are easy to see. the MSB seam above is a real example of a palette creating a boundary out of nothing. + +what a palette cannot do is create the tiling in NAND or the bands in ADD. those survive every palette, including flat monochrome, and both were confirmed by brute force over all pairs rather than by looking. the colour makes them legible. + +## Shifts, and three wires + +8 by 8 left-shift visualization + +*left shift, vertical striping with period 8. [^lshft-link]* + +the shift count is masked to `b & 7`, so $B$ and $B + 8$ shift by the same amount. the whole image repeats every 8 columns of $B$, verified across all pairs. + +this one is worth following into the gates, because it is the shortest path from a picture to a circuit. + +a shift distance in an 8-bit word ranges over 0 to 7, which needs exactly 3 bits. so a barrel shifter takes only the low 3 bits of $B$ and ignores the other 5 entirely. it is built as 3 stages of 2-to-1 multiplexers: stage 0 shifts by 1 or not, stage 1 by 2 or not, stage 2 by 4 or not. any distance from 0 to 7 is a sum of those, which is just the binary expansion of the shift count. + +the visual period of 8 is those 3 wires. widening to 16 bits makes it 4 stages and a period of 16, and nothing else about the shape changes. + +## Multiplication + +back to the opening image. multiplication is the busiest of the operators, and it is far from chaotic. + +structure that is actually there: + +- powers of two are clean lines. multiplying by $2^k$ is a shift, so those rows and columns look like the shift image. +- zero cells are sparse, 1,280 of 65,536, about 2%. these are the pairs whose product is divisible by 256. +- exactly 25% of results are odd, and the reason is immediate: a product is odd only when both operands are odd, which is a quarter of the grid. + +the texture is a multiplication table with a mod-256 fold on top. the divisor structure of 256 does the visible work, which is why powers of two stand out. + +## Division + +8 by 8 division visualization + +*division, mostly empty. [^div-link]* + +the same shape as the ascii table, at 64 times the resolution. -context: +most of the image is empty. the upper-left region is all zero, because that is where $B > A$, and truncating integer division sends every one of those pairs to 0. -playing around the major rework of the campagin of turing complete 2, -shape_of_logic_tc_major.png +the bright diagonal edge is the $A / B = 1$ region, and it widens as the numbers grow. the count of divisors $B$ giving a quotient of exactly 1 is $\lceil A/2 \rceil$: for $A = 8$ there are 4, for $A = 64$ there are 32, for $A = 255$ there are 128. the tolerance for "these two are basically the same size" scales with magnitude. -i workded on the next version of the ALU. +all the interesting variation is crammed into small values near the axes, where the quotient changes fast. -the new ALU includes next to the usual -- one cycle addition, subtraction -- multiplication +division is also the only operator here that throws information away, and that shows up as large flat regions where many inputs collapse onto one output. the other operators are reversible given one argument. division is not, and the picture is where you see it. +## What I found instead -also division. -for which a new circuit is needed. +this started in [Turing Complete](https://store.steampowered.com/app/1444480/Turing_Complete/), building a new ALU during the campaign rework. -i didn't know how to build the circuit so i went searching for pattterns in bits. +i wanted to write it up in late july, when the viewer was fresh. the time to do it properly only showed up now, which at least meant coming back to the images with enough distance to check the claims instead of trusting the first reading. -i started with a simple script that yields an ascii truth table. -Given two inputs A and B as binary strings, the resulting A / B (integer division, mod) +![Turing Complete major update announcement](/assets/blog/shape_of_logic_tc_major.png) -... +the ALU had one-cycle addition, subtraction, and multiplication. i wanted division, did not know how to build the circuit, and went looking for a pattern in the bits that would tell me. -I didn't quite see the pattern yet in 1 and 0s +i found a different thing, and a better one. the pictures explain *why* division resists the treatment the other three accept: those flat collapsing regions and that widening diagonal are the shape of an operation that discards information and cannot be done in one pass of independent bit logic. addition tiles until carry couples it. shift is 3 multiplexer stages you can read off the image. division is neither. -So I tried different unicode values for 1 and zero, leaving zero empty, focusing only on zero... +the shifter section above is the proof that this route reaches gates at all. for division it points at the answer without handing it over, which leaves the circuit for me to build. -then i treid generating images -first black and white. -1 = white -0 = black +

+how the viewer got built, ascii to shaders -this was neat but not quite there yet. +the path here was not direct. -i thought about other ways to color. -what about one pixel per BYTE instead of BIT +it started with the ascii table above, but printed at 8 bits, where it runs 256 rows wide. reading 1s and 0s i saw nothing, and blanking the zeros is what first made the triangle pop. -there we already get more interesting patterns +then images. first pure black and white, one pixel per *bit*, 1 white and 0 black. neat, not informative. -but how to color? -naive is graycscale 0 = black, 255 = white +the first real jump was one pixel per **byte** instead of per bit. that is what makes a 256x256 image cover an entire 8-bit operator, and patterns appeared immediately. grayscale by value helped further, and the per-bit palettes came later. -this already helps. +but the ALU is 16-bit, and 8 bits was not the target: -but we are still not painting the full pictue -the alu is 16 bit. +- 8-bit: $2^8 = 256$, so $256 \times 256 = 65{,}536$ pixels. fine. +- 16-bit: $2^{16} = 65{,}536$, so $65{,}536 \times 65{,}536 = 4{,}294{,}967{,}296$ pixels. at one byte each that is about 4 GB, and generating it eagerly was hopeless. -we need more resolution. +precomputing full images was already too slow well before that. -i tried generating images for full 8x8 bit combinations but this was wayyyb too slow. +the fix was to stop materialising the image at all: render a viewport and compute only the pixels currently on screen. that pushed the work to a fragment shader evaluating the operator per pixel on the GPU, which is also what makes panning and zooming feel like an image rather than a render queue. +i used Codex for the viewer implementation, the shader, the viewport, and the tooltip plumbing. the analysis and the claims in this post are mine, and each numeric claim was checked against a brute-force pass over all 65,536 pairs rather than taken from the picture. -so i turned to an LLM to see if codex could generate a pattern viewer. -we started at image rendering clietn side -we eventually moved to shaders that evaluate logic at level of individual pixels. -viewport sizing, toolitps with color encoded bytes. +Tooltip showing binary, decimal, signed decimal and hex for a cell -8 bits is nice but can we go larger? +*the tooltip, reading out one cell.* -8 bits -2^8, 256x256 = 65'536 pixels, 8 bits per pixel -524'288 bits of raw, not tiny but managable. +it shows binary, decimal, signed decimal and hex, with the per-bit colour strips that explain what you are looking at. it is what makes the images checkable instead of decorative, since you can point at any pixel and see the arithmetic. -2^16 = 65'536x65'536 = 4'294'967'296 pixels -with 8 bits per pixel yields 34'359'738'368 which is simply not managable. -So i thought about ways we could achieve visualizations without reaching over memeory limits. then it hit me. why don't we simply provide a viewport to vierw the image one part at a time? +the 16-bit view still renders, though it shows the same patterns at higher resolution rather than new ones. the structure was already fully present at 8 bits, which in hindsight is the point: these are properties of the operators, not of the width. -The interactive viewport led to shaders. -you cannot download the full image, but you can see an approximation. -it shows more detail than the 8x8 one but shows patterns only marginally better. +
+## Try it -i thought about hey why are we even looking at patterns of division now? -what would NAND gates look like? What about addition? do we see the carry pattern, the overflow. Can we see the chaotic nature of multiplication? +the viewer is live at [cb341.dev/logic-visualizer](https://cb341.dev/logic-visualizer/). drag to pan, wheel to zoom, arrow keys change operator, click any pixel for the readout. -Palletes -- MSB invert (let leading bit determine sign of number, negative numbers are shown inverted ) -- nibble rgb (each nibble = 4bits gets own part of the specturm, patterns are reconglizable not solely on value but in pattern within the number. ) -- vibrant (tested some colors that worked, colors that didn't work) -- monochrome - simple, stupid +![The viewer toolbar: image size, palette, zoom, jump-to-cell and download controls](/assets/blog/shape_of_logic_toolbar.png) -What if we play with color pallettes +*everything is in the toolbar.* +operator, bit width and palette are dropdowns, Jump takes an $A$, an operator and a $B$ and centres that cell, and Download PNG saves what you are looking at. nobody needs to touch a query string. -Now onto division +the links under each image above are just deep links to a view you could equally reach with the controls. if you do want to build one by hand: -{division} +``` +?theme=0&image=8&op=MUL&a=0&b=0 +``` -What you can clearly see is the diagonal stripe -we round results, discard reminder. -the tolerance for identity grows with size +`op` takes a name or an index: `NAND`, `AND`, `NOR`, `OR`, `XOR`, `ADD`, `MUL`, `DIV`, `LSHFT`, `RSHFT`, `CMP`. -most change hapesn with smal vlaues, -regulartiy with odd/even +`theme` selects the palette: -bands +| value | palette | what it does | +| --- | --- | --- | +| `0` | Vibrant | each bit gets a saturated hue, mixed per pixel | +| `1` | Monochrome | grayscale ramp, no hue information | +| `2` | MSB Invert | vibrant, but colours flip when the leading bit is set | +| `3` | Nibble RGB | high and low nibble map to separate parts of the spectrum | +`image` is the bit width and accepts only 4, 8, or 16. -we can see upper right triangle of the matrix is empty. -if we divide x by y with y being larger, we get zero. -this checks out. +## What's left +i still owe the divider circuit, which means sequential logic and more patience than pattern-matching asked for. -NOTES/IDEAS +the pictures were worth it on their own. an operator has a shape, the shape is specific to what the operation does, and you can read hardware off it: carry breaks the tiles, the shifter is 3 stages of multiplexers, the byte wraps 255 times and each wrap is a line you can point at. -- Include MY logic gate cirucits for NAND,NOR,MUL,ADD,SUB +it is very obviously not random. -Didn't quite find the pattern yet but I will once i get back to spending time in the emulator +## Deep links -Another perspective on Logic. -It is definitiely not random +[^mul-link]: -It is beautful +[^nand-link]: +[^add-link]: +[^lshft-link]: -LIVE DEMO -https://cb341.dev/logic-visualizer/ +[^div-link]: diff --git a/assets/blog/shape_of_logic_add_8x8.png b/assets/blog/shape_of_logic_add_8x8.png new 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ztfQ{DtUJsUq@toP38$fe(s;SCFW=uw_Fpp`q)R(t9VK*2?}<4ebTVwGlFw*pB_0Dz ztoW3CVdh`3p6C!wtG6YqA*qdw>b-2ef8 z@Ex9DK(!y!apl{d9h3&dkWVx~Y2e$0>oQ(hh^SocRHnoeSV32_p{X1nFfX#!z?%@2*b zCBU#3kxmXo^m?IAr||Qil{Wu+g1yr5zdRX7&A(diF$6BFob2xp{&|rsZxB4y5T$k0 zy7`-}`d<_D|MH8a^Nq5RpT#xyLjS*>s6_H0_-Y#|>)8E%_x~^d(yxCtZgn}`?@W0A oUWotYBmQG~|GaKRN%1bJVP0BM1k)})IKYqQecgK{Dwe_j1BWD(nE(I) literal 0 HcmV?d00001 From 0b9f520b2492f18e0e34d386ac886fa5cc9e3997 Mon Sep 17 00:00:00 2001 From: cb341 Date: Sun, 6 Sep 2026 16:41:34 +0200 Subject: [PATCH 03/14] Update wording in 67 --- _posts/2026-09-05-comparing-six-and-seven-from-scratch.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/_posts/2026-09-05-comparing-six-and-seven-from-scratch.md b/_posts/2026-09-05-comparing-six-and-seven-from-scratch.md index 5b57892..5376a52 100644 --- a/_posts/2026-09-05-comparing-six-and-seven-from-scratch.md +++ b/_posts/2026-09-05-comparing-six-and-seven-from-scratch.md @@ -185,7 +185,7 @@ worked on $1 + 2 = 3$. the reason column names the rule and how it is used. $(\r $$ \begin{array}{rl} -& \textbf{Rules used} \\[2pt] +& \textbf{Definitions} \\[2pt] \colorbox{#fff3cd}{$\vphantom{Ag}\text{i.}$} & 1 \stackrel{\mathrm{def}}{=} \operatorname{succ}(0) \\[2pt] \colorbox{#cfe2ff}{$\vphantom{Ag}\text{ii.}$} & 2 \stackrel{\mathrm{def}}{=} \operatorname{succ}(1) \\[2pt] \colorbox{#e2d9f3}{$\vphantom{Ag}\text{iii.}$} & 3 \stackrel{\mathrm{def}}{=} \operatorname{succ}(2) \\[2pt] From b4229e03678f9b6a1e3d2a903dd81af2ccad83e6 Mon Sep 17 00:00:00 2001 From: cb341 Date: Sun, 6 Sep 2026 16:56:52 +0200 Subject: [PATCH 04/14] Polish --- _posts/2026-09-06-shape-of-logic.md | 216 ++++++++++++++-------------- 1 file changed, 108 insertions(+), 108 deletions(-) diff --git a/_posts/2026-09-06-shape-of-logic.md b/_posts/2026-09-06-shape-of-logic.md index b0fe20f..3ab3e81 100644 --- a/_posts/2026-09-06-shape-of-logic.md +++ b/_posts/2026-09-06-shape-of-logic.md @@ -1,97 +1,96 @@ --- title: "Shape of Logic" -description: "Every byte pair of an 8-bit operator fits in one 256x256 image. The pictures show tiling, carry, and overflow as shapes." +description: "Every input pair of an 8-bit operator fits in one 256x256 image. Bitwise ops tile by scale, addition tiles by translation, and overflow turns out to be 255 evenly spaced lines." tags: ["logic", "visualization", "hardware", "first principles"] math: true --- -8 by 8 multiplication visualization +an 8-bit operator takes two bytes and returns one. its truth table has $256 \times 256 = 65{,}536$ rows, which is few enough to put every one of them on screen at the same time. -*multiplication, every pair of bytes. [^mul-link]* +256 by 256 multiplication truth table -what is this? +*multiplication, every input pair at once. [open in the viewer](https://cb341.dev/logic-visualizer/?theme=0&image=8&op=MUL)* ## The idea -an 8-bit operator takes two bytes and returns one. that is $256 \times 256 = 65{,}536$ input pairs, and the whole thing fits on a screen. - put $A$ on the x axis, $B$ on the y axis, and colour the pixel at $(A, B)$ by the result of $A \mathbin{\mathrm{op}} B$. the origin sits bottom-left, so $A$ grows rightwards and $B$ upwards. -that image is the operator's truth table over that domain, every input pair at once. +that image is the operator's whole truth table over that domain. -the colour is per-bit rather than a value ramp: each of the 8 result bits owns a hue, and a pixel mixes the hues of the bits that are set. that matters later, because two numerically close values can look nothing alike. +the colour is per-bit rather than a value ramp. each of the 8 result bits owns a hue and a pixel mixes the hues of the bits that are set, so two numerically close values can look nothing alike. -## Starting in ascii: 4-bit division +## Starting in ASCII: 4-bit division -the first version printed characters. this is division, $A / B$, at 4 bits, with set bits as `@` and clear bits blank. +the first version printed characters. this is integer division, $A // B$, at 4 bits, set bits as `@` and clear bits blank. $A$ runs across, $B$ runs up, same as every image here. ``` - A\B 0000 0001 0010 0011 0100 0101 0110 0111 1000 1001 1010 1011 1100 1101 1110 1111 + A 0000 0001 0010 0011 0100 0101 0110 0111 1000 1001 1010 1011 1100 1101 1110 1111 + B + 1111 @ + 1110 @ @ + 1101 @ @ @ + 1100 @ @ @ @ + 1011 @ @ @ @ @ + 1010 @ @ @ @ @ @ + 1001 @ @ @ @ @ @ @ + 1000 @ @ @ @ @ @ @ @ + 0111 @ @ @ @ @ @ @ @ @ + 0110 @ @ @ @ @ @ @ @ @ @ + 0101 @ @ @ @ @ @ @ @ @ @ @@ + 0100 @ @ @ @ @ @ @ @ @@ @@ @@ @@ + 0011 @ @ @ @ @ @ @@ @@ @@ @ @ @ @ @ + 0010 @ @ @ @ @@ @@ @ @ @ @ @ @ @@ @@ @@@ @@@ + 0001 @ @ @@ @ @ @ @@ @@@ @ @ @ @ @ @ @@ @@ @@ @ @@@ @@@@ 0000 - 0001 @ - 0010 @ @ - 0011 @@ @ @ - 0100 @ @ @ @ - 0101 @ @ @ @ @ @ - 0110 @@ @@ @ @ @ @ - 0111 @@@ @@ @ @ @ @ @ - 1000 @ @ @ @ @ @ @ @ - 1001 @ @ @ @@ @ @ @ @ @ @ - 1010 @ @ @ @ @@ @ @ @ @ @ @ @ - 1011 @ @@ @ @ @@ @ @ @ @ @ @ @ @ - 1100 @@ @@ @ @@ @ @ @ @ @ @ @ @ - 1101 @@ @ @@ @ @@ @ @ @ @ @ @ @ @ @ - 1110 @@@ @@@ @ @@ @ @ @ @ @ @ @ @ @ @ - 1111 @@@@ @@@ @ @ @@ @@ @ @ @ @ @ @ @ @ @ @ ``` -leaving the zeros blank is what makes the shape appear. the empty upper-right region is already there, and so is the staircase running down the diagonal. +blanking the zeros is what makes the shape appear. the empty upper-left region and the staircase along the diagonal are both already here. -at 4 bits this is readable. at 8 bits it is 256 rows of 256 columns and the terminal stops being the right tool. +at 4 bits this fits on a screen. at 8 bits it is 256 rows of 256 columns and the terminal stops being the right tool. ## Bitwise operators tile -8 by 8 NAND visualization +256 by 256 NAND truth table -*NAND, monochrome. [^nand-link]* +*NAND, monochrome. [open in the viewer](https://cb341.dev/logic-visualizer/?theme=1&image=8&op=NAND)* the picture is made of nested copies of itself. a quadrant repeats the whole, and inside it the same block repeats again, down to single pixels. -the reason is that bitwise operators have no communication between bit positions. result bit $k$ depends on $A_k$ and $B_k$ and nothing else. so the 8-bit table is four copies of the 7-bit table, which is four copies of the 6-bit table, all the way down. +each bit position is evaluated independently of the others. result bit $k$ is a function of $A_k$ and $B_k$ alone, so the 8-bit table is four copies of the 7-bit table, which is four copies of the 6-bit table, all the way down. -the self-similarity goes as deep as the bit count allows. for all 65,536 pairs, the low nibble of $A \mathbin{\overline{\wedge}} B$ depends only on the low nibbles of $A$ and $B$. +for all 65,536 pairs, the low nibble of $A \mathbin{\overline{\wedge}} B$ depends only on the low nibbles of $A$ and $B$. the self-similarity goes as deep as the bit count allows. -## Addition breaks the tiling +## Addition tiles diagonally -8 by 8 addition visualization +addition truth table in nibble RGB -*addition, nibble RGB. the blocks are gone. [^add-link]* +*addition at 16 bits, nibble RGB. blocks become bands. [open in the viewer](https://cb341.dev/logic-visualizer/?theme=3&image=16&op=ADD)* -carry. bit $k$ of $A + B$ depends on every bit below $k$. the columns are coupled, the independence is gone, and the nested blocks go with it. the same nibble test that holds for NAND fails for addition, exactly as the carry chain predicts. +carry. bit $k$ of $A + B$ depends on every bit below $k$, so the bits are no longer independent and the nested blocks go with them. the nibble test that passes for NAND fails here, because the low nibble of a sum can be changed by a carry out of it. -what replaces the blocks is diagonal bands. $A + B$ is constant along an anti-diagonal, so each band is one result value. +the repetition does not stop, it changes direction. $A + B$ is constant along an anti-diagonal, so the pattern is bands instead of blocks. it is still a tiling. slide the image one step right and one step down and it lands on itself exactly. -this is the seam where logic stops and arithmetic begins, and it is visible. +the difference is what kind of repetition you get. NAND repeats by scale, the same block nested at every size. addition repeats by translation, one motif shifted along a diagonal. carry is what turns one into the other. the palette here splits the result by nibble, so the high nibble drives blue and the low nibble drives red. the leading bit lives in that high nibble, which makes the cool bands the results a signed reading calls negative, and the warm ones positive. -read along a diagonal and the bands alternate. crossing one edge flips the sign: two positive numbers add to a negative one, two negatives add to a positive. that is signed overflow, and it is not rare. it happens in exactly 25% of all pairs, 8,128 cells where positive plus positive goes negative and 8,256 where negative plus negative goes positive. +the bands alternate, and crossing an edge flips the sign. two positive numbers add to a negative one, two negatives add to a positive. that is signed overflow, and it covers a quarter of the grid. the rate is the same 25% at 8 bits and at 16, split almost evenly between the two directions. -the fine striping is because this is the 16-bit view. the bands survive the widening, which is the first hint that these are properties of the operator rather than of the byte. +the fine striping is because this is the 16-bit view. the bands survive the widening. ## Overflow has a shape -there are 256 bands, and each one is a residue class $[r]_{256}$. counting cells, every class holds exactly 256 of the 65,536 pixels, uniformly. +there are 256 bands, one per result value, and each holds exactly 256 of the 65,536 pixels. -the bands are the cosets of $256\mathbb{Z}$ in $\mathbb{Z}$, drawn to scale. +a band is every pair that sums to the same byte. the bands partition the grid, and that partition is addition mod 256 drawn to scale. -take $148 + 138$. the true sum is $286$. the byte says $30$. +take $148 + 138$. the true sum is $286$. the byte is $30$. -$$286 \equiv 30 \pmod{256}$$ +$$148 + 138 \equiv_{256} 30$$ that wrap is a band edge. every place the diagonal stripe restarts, an overflow just happened, and there are 255 such edges at even spacing across the image. -overflow is usually taught as a bug class, a thing that bites you at the boundary. the picture gives it a shape: $\mathbb{Z}/256\mathbb{Z}$, the quotient, made visible. the boundary is not an edge case sitting at the end of the range. it is a wall you hit once per band, over and over. +i had thought of overflow as something that happens at the end of the range, once, when you run out of room. there are 255 band edges spread evenly across the image and each one is an overflow. it happens everywhere, at regular intervals, not at a boundary. ## The seam that isn't there @@ -99,86 +98,88 @@ if two's complement were a property of the data, there would be a visible seam a there is none. across all 65,536 pairs the result bits of signed and unsigned addition are identical everywhere. the same holds for subtraction and for multiplication. -this is the whole trick of two's complement. it is a reading convention applied to bits. the hardware adder does not know which convention you are using, and that is exactly why the representation won. - -the image cannot show a distinction that is not in the data. - -you can make a seam appear by choosing the MSB-invert palette, which flips the colour when the leading bit is set. that seam lives in the palette. +this is the trick of two's complement. it is a reading convention applied to bits, and the adder does not know which convention you are using. one adder serves both, which is why the representation won. -## Owning the rendering +you can make a seam appear by picking the MSB-invert palette, which flips the colour when the leading bit is set. that seam is in the palette. -a colour map can invent structure that the data does not have, so this deserves an answer. +## Is it the palette or the data -the palette is a per-bit layering, and switching it changes which patterns are easy to see. the MSB seam above is a real example of a palette creating a boundary out of nothing. +a colour map can invent structure that the data does not have. the MSB seam above is exactly that, a boundary that exists because the palette draws one. -what a palette cannot do is create the tiling in NAND or the bands in ADD. those survive every palette, including flat monochrome, and both were confirmed by brute force over all pairs rather than by looking. the colour makes them legible. +the tiling in NAND and the bands in ADD show up under all four palettes, monochrome included, and i checked both by brute force over all pairs instead of trusting the picture. the colour makes them easier to see and does not put them there. ## Shifts, and three wires -8 by 8 left-shift visualization +256 by 256 left-shift truth table -*left shift, vertical striping with period 8. [^lshft-link]* +*left shift, vertical striping with period 8. [open in the viewer](https://cb341.dev/logic-visualizer/?theme=2&image=8&op=LSHFT)* the shift count is masked to `b & 7`, so $B$ and $B + 8$ shift by the same amount. the whole image repeats every 8 columns of $B$, verified across all pairs. -this one is worth following into the gates, because it is the shortest path from a picture to a circuit. +this is the one place where the picture hands you a circuit. -a shift distance in an 8-bit word ranges over 0 to 7, which needs exactly 3 bits. so a barrel shifter takes only the low 3 bits of $B$ and ignores the other 5 entirely. it is built as 3 stages of 2-to-1 multiplexers: stage 0 shifts by 1 or not, stage 1 by 2 or not, stage 2 by 4 or not. any distance from 0 to 7 is a sum of those, which is just the binary expansion of the shift count. +a shift distance in an 8-bit word runs 0 to 7, which needs 3 bits. the shifter takes the low 3 bits of $B$ and ignores the other 5. it is 3 stages of 2-to-1 multiplexers, shifting by 1, by 2, by 4, each stage on or off. any distance from 0 to 7 is a sum of those three, the binary expansion of the shift count. -the visual period of 8 is those 3 wires. widening to 16 bits makes it 4 stages and a period of 16, and nothing else about the shape changes. +the period of 8 in the image is those 3 wires. at 16 bits it becomes 4 stages and a period of 16. ## Multiplication -back to the opening image. multiplication is the busiest of the operators, and it is far from chaotic. - -structure that is actually there: +back to the opening image. multiplication is the busiest operator here, and every part of that texture has a cause. - powers of two are clean lines. multiplying by $2^k$ is a shift, so those rows and columns look like the shift image. - zero cells are sparse, 1,280 of 65,536, about 2%. these are the pairs whose product is divisible by 256. -- exactly 25% of results are odd, and the reason is immediate: a product is odd only when both operands are odd, which is a quarter of the grid. +- exactly 25% of results are odd. a product is odd only when both operands are odd, which is a quarter of the grid. + +underneath all of it is one rule. bit $k$ of $A \times B$ depends only on the low $k+1$ bits of both operands, which i checked for every bit and every pair. carry moves information upward and never downward, so the low bits of a product never learn about the high bits of its inputs. -the texture is a multiplication table with a mod-256 fold on top. the divisor structure of 256 does the visible work, which is why powers of two stand out. +the layout follows from that. the low nibble of the product is fixed by the low nibbles of the inputs alone, which draws the 16x16 grid of cells repeating their fine detail. the high bits depend on everything, so they vary slowly and paint the 4x4 arrangement of large squares on top. the nibble grid you can see is that split between fast and slow bits. + +the curves inside each cell are level sets of $A \times B$, which are hyperbolas. before the modulo they would be single arcs sweeping from axis to axis. the wrap at 256 chops each one into pieces, and the pieces stack into the nested rings that read as fish scales. + +their spacing follows from the gradient. $A \times B$ changes at rate $\sqrt{A^2 + B^2}$, so arcs sit roughly $256 / \sqrt{A^2 + B^2}$ pixels apart. near the origin that is 22 pixels and the arcs are lazy and wide. at $A = B = 128$ it is 1.4 pixels. past $A = B = 181$ it drops below one pixel, the arcs stop being resolvable, and the corners taper into flat interference where the true product wraps 254 times before it lands. + +the vortex look comes from that spacing gradient bending families of arcs against each other. it is a moiré pattern, an artifact of sampling curves too finely for the pixel grid, though the arithmetic underneath really is moving that fast. ## Division -8 by 8 division visualization +256 by 256 division truth table -*division, mostly empty. [^div-link]* +*division, mostly empty. [open in the viewer](https://cb341.dev/logic-visualizer/?theme=0&image=8&op=DIV)* -the same shape as the ascii table, at 64 times the resolution. +the same shape as the ASCII table, at 64 times the resolution. most of the image is empty. the upper-left region is all zero, because that is where $B > A$, and truncating integer division sends every one of those pairs to 0. -the bright diagonal edge is the $A / B = 1$ region, and it widens as the numbers grow. the count of divisors $B$ giving a quotient of exactly 1 is $\lceil A/2 \rceil$: for $A = 8$ there are 4, for $A = 64$ there are 32, for $A = 255$ there are 128. the tolerance for "these two are basically the same size" scales with magnitude. +the bright diagonal edge is the $A // B = 1$ region, and it widens as the numbers grow. the count of divisors $B$ giving a quotient of exactly 1 is $\lceil A/2 \rceil$: for $A = 8$ there are 4, for $A = 64$ there are 32, for $A = 255$ there are 128. the tolerance for "these two are basically the same size" scales with magnitude. all the interesting variation is crammed into small values near the axes, where the quotient changes fast. -division is also the only operator here that throws information away, and that shows up as large flat regions where many inputs collapse onto one output. the other operators are reversible given one argument. division is not, and the picture is where you see it. +division is the only operator here that throws information away. that is what the large flat regions are, many inputs landing on the same output. given one argument the others are reversible and division is not. ## What I found instead -this started in [Turing Complete](https://store.steampowered.com/app/1444480/Turing_Complete/), building a new ALU during the campaign rework. +this started in [turing complete 2.0](https://store.steampowered.com/app/1444480/Turing_Complete/), building a new ALU during the campaign rework. -i wanted to write it up in late july, when the viewer was fresh. the time to do it properly only showed up now, which at least meant coming back to the images with enough distance to check the claims instead of trusting the first reading. +i wanted to write it up in late july. the time only showed up now, which meant coming back to the images cold and checking the claims instead of trusting what i remembered seeing. -![Turing Complete major update announcement](/assets/blog/shape_of_logic_tc_major.png) +Turing Complete major update announcement -the ALU had one-cycle addition, subtraction, and multiplication. i wanted division, did not know how to build the circuit, and went looking for a pattern in the bits that would tell me. +the ALU had one-cycle addition, subtraction, and multiplication. i wanted division on the same terms, a dedicated circuit that answers in one pass instead of a loop that shifts and subtracts once per bit. the usual restoring divider costs a cycle per output bit, which for 16 bits is 16 cycles against multiplication's one. -i found a different thing, and a better one. the pictures explain *why* division resists the treatment the other three accept: those flat collapsing regions and that widening diagonal are the shape of an operation that discards information and cannot be done in one pass of independent bit logic. addition tiles until carry couples it. shift is 3 multiplexer stages you can read off the image. division is neither. +those first three i could reason my way to. division i could not, and it did not become obvious no matter how long i stared at the spec, so i went looking for a pattern in the bits instead. if the truth table had visible structure, maybe the structure was a circuit. -the shifter section above is the proof that this route reaches gates at all. for division it points at the answer without handing it over, which leaves the circuit for me to build. +i did not get a circuit out of it. what i got is why division resists the treatment the other three accept. the flat collapsing regions and the widening diagonal are an operation that discards information, and that cannot come from one pass of independent bit logic. addition tiles until carry couples it. shift is 3 multiplexer stages you can read straight off the image. division is neither. - +## One more, at 16 bits + +everything above is 8 bits, a 256x256 image that fits on a page. the viewer also runs at 16, and that is a different kind of object. + +a 16-bit operator has $65{,}536 \times 65{,}536$ input pairs. that is 4,294,967,296 of them, every one drawn as a pixel, in a single image 65,536 pixels on a side. as raw RGB it would be 12 GB. nothing hands you a 12 GB image, so the viewer never builds one, and the shader computes only the pixels the window is showing. + +so this is a full 16-bit operation, every input pair defined, and you are looking at a piece of it. + +A 16-bit operator, showing nested curved arcs + +*which one?* + ## Try it the viewer is live at 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zx_!paJ?CmDrT^t5MKgriQ;f)|#$|1ZD3Mrj6<8YO6q9K{1=t{)+@!>jIsyk4FoaTPH#P(a>3E1v5o6~pb)X_Ow& z7a&ZExgZem1GoS?=5rDh%wQ@jzRr5&e*+x6M3X-O+){7=QJQGUm&LZBKSp-_-e2EV z%BN&fU;z~RQ@=eo#cGvcnKd~A@rpCPN5PPq>;QBPAhL}X{zL$fYuuOOAi(Jgb7mLN@Ok6I>LfS-~?>U7rg;M0vSZwa(LcmS~=w2tB(Owa*G zU@}n^RE;&Cf~Nna3*%5}pK>`&nykyUp__}PpDO!CbCblX&%VF6%apraaRDL8Fjy@x zZFs0egPy&+keYEml!|UHFM9M1;TXXP$bwTNYaM1f=FVE)3&8F{PXeIPV1qQPrQyK| z+lh+C!EmvsITZ;z`=fe9E+$gU=joejnp}3GDg}Tt4*@+AnZx=%A%F);MM?{_%4x6{MJ^mXyG%)m0=GAY+r@wQ5g>nfX zSk3F~=OvevNfZN<2x@qZd}b`>b7%rfXy!wVff?n+?{GVZPpEW$tzpSnR` z<=VSI(nbWV%iJzkLx59#H-IGu=lBv3#vhItn*>tvN)+(8$F_WhIzUlQX+3jby`=G{ z?689Joe}~*WF#y(u|}$3O%MT0GQt4~7&^8$tP(P1srTr03UfSxZSNglj&5EM0}gO$ z+IIVcx=rQJb#_QX!RR5&0j!}e9tM6}PPRQZ_$A(HG9QZ2A8DWLv$ Date: Sun, 6 Sep 2026 17:53:20 +0200 Subject: [PATCH 10/14] Update --- _posts/2026-09-06-shape-of-logic.md | 18 ++++-------------- 1 file changed, 4 insertions(+), 14 deletions(-) diff --git a/_posts/2026-09-06-shape-of-logic.md b/_posts/2026-09-06-shape-of-logic.md index 0badb66..06b9d79 100644 --- a/_posts/2026-09-06-shape-of-logic.md +++ b/_posts/2026-09-06-shape-of-logic.md @@ -184,8 +184,7 @@ i did not get a circuit out of it. what i got is why division resists the treatm so the method reaches gates, the shifter proves that much. for division it stops short and i still have to build the thing myself. -

-how the viewer got built, ASCII to shaders +### How the viewer got built, ASCII to shaders it started with the ASCII table above, but printed at 8 bits, where it runs 256 rows wide. reading 1s and 0s i saw nothing, and blanking the zeros is what first made the triangle pop. @@ -193,12 +192,7 @@ then images. black and white, one pixel per *bit*, 1 white and 0 black. neat, no what changed it was one pixel per byte instead of per bit. that is what makes a 256x256 image cover an entire 8-bit operator, and patterns showed up immediately. grayscale by value helped, and the per-bit palettes came later. -but the ALU is 16-bit, and 8 bits was not the target: - -- 8-bit: $2^8 = 256$, so $256 \times 256 = 65{,}536$ pixels. fine. -- 16-bit: $2^{16} = 65{,}536$, so $65{,}536 \times 65{,}536 = 4{,}294{,}967{,}296$ pixels. at one byte each that is about 4 GB, and generating it eagerly was hopeless. - -precomputing full images was already too slow well before that. +256x256 was fine to generate ahead of time. the ALU is 16-bit though, and precomputing at that width was hopeless well before i worked out how hopeless. the fix was to stop materialising the image at all. render a viewport, compute only the pixels currently on screen. that pushed the work to a fragment shader evaluating the operator per pixel on the GPU, which is also why panning and zooming feel like an image rather than a render queue. @@ -210,21 +204,17 @@ i used Codex for the viewer implementation, the shader, the viewport, and the to it shows binary, decimal, signed decimal and hex, with the per-bit colour strips. point at any pixel and you get the arithmetic for that cell, which is how i checked the readings above. -the 16-bit view renders the same way, and the patterns hold up rather than dissolving. the operators keep their character at both widths. - -
- ## One more, at 16 bits everything above is 8 bits, a 256x256 image that fits on a page. the viewer also runs at 16, and that is a different kind of object. -a 16-bit operator has $65{,}536 \times 65{,}536$ input pairs. that is 4,294,967,296 of them, every one drawn as a pixel, in a single image 65,536 pixels on a side. as raw RGB it would be 12 GB. nothing hands you a 12 GB image, so the viewer never builds one, and the shader computes only the pixels the window is showing. +a 16-bit operator has $65{,}536 \times 65{,}536$ input pairs. that is 4,294,967,296 of them, one image 65,536 pixels on a side. the results alone, two bytes each, are 8 GB. this is why nothing is precomputed and the shader draws only what the window covers. so this is a full 16-bit operation, every input pair defined, and you are looking at a piece of it. A 16-bit operator, showing nested curved arcs -*which one?* +*16x16, which one?* ## Try it From e747393d2ca025c73590d9bff658272174964440 Mon Sep 17 00:00:00 2001 From: cb341 Date: Sun, 6 Sep 2026 18:07:10 +0200 Subject: [PATCH 11/14] Polish --- _posts/2026-09-06-shape-of-logic.md | 31 +++++++++++++++-------------- 1 file changed, 16 insertions(+), 15 deletions(-) diff --git a/_posts/2026-09-06-shape-of-logic.md b/_posts/2026-09-06-shape-of-logic.md index 06b9d79..50cf5ed 100644 --- a/_posts/2026-09-06-shape-of-logic.md +++ b/_posts/2026-09-06-shape-of-logic.md @@ -56,27 +56,27 @@ at 4 bits this fits on a screen. at 8 bits it is 256 rows of 256 columns and the the picture is made of nested copies of itself. a quadrant repeats the whole, and inside it the same block repeats again, down to single pixels. -each bit position is evaluated independently of the others. result bit $k$ is a function of $A_k$ and $B_k$ alone, so the 8-bit table is four copies of the 7-bit table, which is four copies of the 6-bit table, all the way down. +each bit position is evaluated independently of the others. result bit $k$ is a function of $A_k$ and $B_k$ alone, so the four quadrants share the same 7-bit pattern in their low bits, and each of those splits the same way again, down to single pixels. only the top bit distinguishes one quadrant from another. for all 65,536 pairs, the low nibble of $A \mathbin{\overline{\wedge}} B$ depends only on the low nibbles of $A$ and $B$. the self-similarity goes as deep as the bit count allows. ## Addition tiles diagonally -addition truth table in nibble RGB +256 by 256 addition truth table in MSB invert -*addition at 8 bits, nibble RGB. blocks become bands. [open in the viewer](https://cb341.dev/logic-visualizer/?theme=2&image=8&op=ADD)* +*addition at 8 bits, MSB invert. blocks become bands. [open in the viewer](https://cb341.dev/logic-visualizer/?theme=2&image=8&op=ADD)* -carry. bit $k$ of $A + B$ depends on every bit below $k$, so the bits are no longer independent and the nested blocks go with them. the nibble test that passes for NAND fails here, because the low nibble of a sum can be changed by a carry out of it. +carry. bit $k$ of $A + B$ depends on every bit below $k$, so the bits are no longer independent and the nested blocks go with them. + +addition still passes the nibble test, for the record. the low nibble of a sum is fixed by the low nibbles of the inputs, because a carry only ever moves upward. what breaks is the stricter property NAND has, where bit $k$ needs nothing but bit $k$. the repetition does not stop, it changes direction. $A + B$ is constant along an anti-diagonal, so the pattern is bands instead of blocks. it is still a tiling. slide the image one step right and one step down and it lands on itself exactly. the difference is what kind of repetition you get. NAND repeats by scale, the same block nested at every size. addition repeats by translation, one motif shifted along a diagonal. carry is what turns one into the other. -the palette here splits the result by nibble, so the high nibble drives blue and the low nibble drives red. the leading bit lives in that high nibble, which makes the cool bands the results a signed reading calls negative, and the warm ones positive. - -the bands alternate, and crossing an edge flips the sign. two positive numbers add to a negative one, two negatives add to a positive. that is signed overflow, and it covers a quarter of the grid. the rate is the same 25% at 8 bits and at 16, split almost evenly between the two directions. +the palette here inverts every colour when the leading bit is set, so the cool bands are the results a signed reading calls negative and the warm ones positive. -the fine striping is because this is the 16-bit view. the bands survive the widening. +the bands alternate, and crossing an edge flips the sign. two positive numbers add to a negative one, two negatives add to a positive. that is signed overflow, and it covers a quarter of the grid, the same 25% at 8 bits and at 16. ## Overflow has a shape @@ -88,9 +88,9 @@ take $148 + 138$. the true sum is $286$. the byte is $30$. $$148 + 138 \equiv_{256} 30$$ -that wrap is a band edge. every place the diagonal stripe restarts, an overflow just happened, and there are 255 such edges at even spacing across the image. +that wrap is where the colour restarts. there are 255 wrap transitions, one in every nonzero row, and together they line up into the diagonal seam running across the image. -i had thought of overflow as something that happens at the end of the range, once, when you run out of room. there are 255 band edges spread evenly across the image and each one is an overflow. it happens everywhere, at regular intervals, not at a boundary. +i had thought of overflow as something that happens at the end of the range, once, when you run out of room. it is a seam through the middle of the table, and every row crosses it. ## The seam that isn't there @@ -98,7 +98,7 @@ if two's complement were a property of the data, there would be a visible seam a there is none. across all 65,536 pairs the result bits of signed and unsigned addition are identical everywhere. the same holds for subtraction and for multiplication. -this is the trick of two's complement. it is a reading convention applied to bits, and the adder does not know which convention you are using. one adder serves both, which is why the representation won. +this is the trick of two's complement. it is a reading convention applied to bits, and the adder does not know which convention you are using. one adder serves both, which is one reason the representation is so hardware friendly. you can make a seam appear by picking the MSB-invert palette, which flips the colour when the leading bit is set. that seam is in the palette. @@ -166,7 +166,7 @@ the bright diagonal edge is the $A // B = 1$ region, and it widens as the number all the interesting variation is crammed into small values near the axes, where the quotient changes fast. -division is the only operator here that throws information away. that is what the large flat regions are, many inputs landing on the same output. given one argument the others are reversible and division is not. +plenty of these operators are many-to-one, AND and the shifts included, so losing information is not what makes division unusual. what is unusual is how much of the table is constant, and how far the quotient reaches for its inputs. the bitwise operators answer bit by bit. division needs the whole magnitude of both arguments before it can say anything. ## What I found instead @@ -180,7 +180,9 @@ the ALU had one-cycle addition, subtraction, and multiplication. i wanted divisi those first three i could reason my way to. division i could not, and it did not become obvious no matter how long i stared at the spec, so i went looking for a pattern in the bits instead. if the truth table had visible structure, maybe the structure was a circuit. -i did not get a circuit out of it. what i got is why division resists the treatment the other three accept. the flat collapsing regions and the widening diagonal are an operation that discards information, and that cannot come from one pass of independent bit logic. addition tiles until carry couples it. shift is 3 multiplexer stages you can read straight off the image. division is neither. +i did not get a circuit out of it. what i got is a clearer sense of why division resists what the other three accept. NAND separates into 8 independent one-bit problems. addition into a carry chain. shift into 3 multiplexer stages you can read straight off the image. division has no such decomposition on show, and its picture is the one that does not break into parts. + +that is not a proof that a fast divider cannot exist. combinational dividers do exist, unrolling the shift-and-subtract loop into hardware and paying in area and propagation depth instead of cycles. the picture says the structure is not separable in the way the others are, and it does not say what the circuit costs. so the method reaches gates, the shifter proves that much. for division it stops short and i still have to build the thing myself. @@ -237,8 +239,7 @@ the operators, in order: | `2` | `NOR` | not (A or B) | | `3` | `OR` | A or B | | `4` | `XOR` | A xor B | -| `5` | `ADDA` | signed addition, same bits as `ADD` | -| `6` | `ADD` | unsigned addition | +| `5`, `6` | `ADD` | addition. indices 5 and 6 label it signed and unsigned, and produce identical bits | | `7` | `MUL` | multiplication, low bits | | `8` | `DIV` | truncating division, B of 0 gives 0 | | `9` | `LSHFT` | A shifted left by the low bits of B | From 0bf5df4a7e8de5639272e393b16f7df56487bd58 Mon Sep 17 00:00:00 2001 From: cb341 Date: Sun, 6 Sep 2026 18:13:47 +0200 Subject: [PATCH 12/14] Add link to table, update URLs --- _posts/2026-09-06-shape-of-logic.md | 6 +++++- 1 file changed, 5 insertions(+), 1 deletion(-) diff --git a/_posts/2026-09-06-shape-of-logic.md b/_posts/2026-09-06-shape-of-logic.md index 50cf5ed..384e9f5 100644 --- a/_posts/2026-09-06-shape-of-logic.md +++ b/_posts/2026-09-06-shape-of-logic.md @@ -110,6 +110,8 @@ it cuts the other way too. nibble RGB puts 256 result values onto 121 colours, a so the palette decides which scale you can see. what it cannot do is put structure into data that has none. the tiling in NAND and the bands in ADD show up under all four palettes, monochrome included, and i checked both by brute force over all pairs instead of trusting the picture. +[every operator in all four palettes](https://cb341.dev/logic-visualizer/table) is one page if you want to compare them yourself. + ## Shifts, and three wires 256 by 256 left-shift truth table @@ -156,7 +158,7 @@ neither picture is more correct. vibrant resolves individual results and buries 256 by 256 division truth table -*division, mostly empty. [open in the viewer](https://cb341.dev/logic-visualizer/?theme=0&image=8&op=DIV)* +*division, mostly empty. [open in the viewer](https://cb341.dev/logic-visualizer/?theme=2&image=8&op=DIV)* the same shape as the ASCII table, at 64 times the resolution. @@ -230,6 +232,8 @@ operator, bit width and palette are dropdowns, Jump takes an $A$, an operator an the link under each image opens that exact view. +[cb341.dev/logic-visualizer/table](https://cb341.dev/logic-visualizer/table) is the contact sheet: every operator at 8 bits, in all four palettes, side by side. it is the fastest way to see how much the palette decides what you notice. + the operators, in order: | index | `op` | operation | From 4dc005a82f49e04a47dccce38b8f4b8cc7f676d3 Mon Sep 17 00:00:00 2001 From: cb341 Date: Sun, 6 Sep 2026 18:22:22 +0200 Subject: [PATCH 13/14] Wrap images --- _posts/2026-09-06-shape-of-logic.md | 16 ++++++++-------- 1 file changed, 8 insertions(+), 8 deletions(-) diff --git a/_posts/2026-09-06-shape-of-logic.md b/_posts/2026-09-06-shape-of-logic.md index 384e9f5..d7cd60e 100644 --- a/_posts/2026-09-06-shape-of-logic.md +++ b/_posts/2026-09-06-shape-of-logic.md @@ -7,7 +7,7 @@ math: true an 8-bit operator takes two bytes and returns one. its truth table has $256 \times 256 = 65{,}536$ rows, which is few enough to put every one of them on screen at the same time. -256 by 256 multiplication truth table +
256 by 256 multiplication truth table *multiplication, every input pair at once. [open in the viewer](https://cb341.dev/logic-visualizer/?theme=2&image=8&op=MUL)* @@ -50,7 +50,7 @@ at 4 bits this fits on a screen. at 8 bits it is 256 rows of 256 columns and the ## Bitwise operators tile -256 by 256 NAND truth table +256 by 256 NAND truth table *NAND, monochrome. [open in the viewer](https://cb341.dev/logic-visualizer/?theme=1&image=8&op=NAND)* @@ -62,7 +62,7 @@ for all 65,536 pairs, the low nibble of $A \mathbin{\overline{\wedge}} B$ depend ## Addition tiles diagonally -256 by 256 addition truth table in MSB invert +256 by 256 addition truth table in MSB invert *addition at 8 bits, MSB invert. blocks become bands. [open in the viewer](https://cb341.dev/logic-visualizer/?theme=2&image=8&op=ADD)* @@ -114,7 +114,7 @@ so the palette decides which scale you can see. what it cannot do is put structu ## Shifts, and three wires -256 by 256 left-shift truth table +256 by 256 left-shift truth table *left shift, vertical striping with period 8. [open in the viewer](https://cb341.dev/logic-visualizer/?theme=2&image=8&op=LSHFT)* @@ -144,7 +144,7 @@ they are wide near the origin, where products grow slowly, and tighten as the nu the same operator in nibble RGB makes that easier to see. -256 by 256 multiplication truth table in nibble RGB +256 by 256 multiplication truth table in nibble RGB *multiplication, nibble RGB. [open in the viewer](https://cb341.dev/logic-visualizer/?theme=3&image=8&op=MUL)* @@ -156,7 +156,7 @@ neither picture is more correct. vibrant resolves individual results and buries ## Division -256 by 256 division truth table +256 by 256 division truth table *division, mostly empty. [open in the viewer](https://cb341.dev/logic-visualizer/?theme=2&image=8&op=DIV)* @@ -176,7 +176,7 @@ this started in [turing complete 2.0](https://store.steampowered.com/app/1444480 i wanted to write it up in late july. the time only showed up now, which meant coming back to the images cold and checking the claims instead of trusting what i remembered seeing. -Turing Complete major update announcement +Turing Complete major update announcement the ALU had one-cycle addition, subtraction, and multiplication. i wanted division on the same terms, a dedicated circuit that answers in one pass instead of a loop that shifts and subtracts once per bit. the usual restoring divider costs a cycle per output bit, which for 16 bits is 16 cycles against multiplication's one. @@ -224,7 +224,7 @@ so this is a full 16-bit operation, every input pair defined, and you are lookin the viewer is live at [cb341.dev/logic-visualizer](https://cb341.dev/logic-visualizer/). drag to pan, wheel to zoom, arrow keys change operator, click any pixel for the readout. -The viewer toolbar: image size, palette, zoom, jump-to-cell and download controls +The viewer toolbar: image size, palette, zoom, jump-to-cell and download controls *everything is in the toolbar.* From 6569db2cf59b5b879c09c336619b5ccdb0b75d4c Mon Sep 17 00:00:00 2001 From: cb341 Date: Sun, 6 Sep 2026 18:29:25 +0200 Subject: [PATCH 14/14] Phrasing, final --- _posts/2026-09-06-shape-of-logic.md | 32 ++++++++++++++--------------- 1 file changed, 15 insertions(+), 17 deletions(-) diff --git a/_posts/2026-09-06-shape-of-logic.md b/_posts/2026-09-06-shape-of-logic.md index d7cd60e..81b1afa 100644 --- a/_posts/2026-09-06-shape-of-logic.md +++ b/_posts/2026-09-06-shape-of-logic.md @@ -228,27 +228,25 @@ the viewer is live at [cb341.dev/logic-visualizer](https://cb341.dev/logic-visua *everything is in the toolbar.* -operator, bit width and palette are dropdowns, Jump takes an $A$, an operator and a $B$ and centres that cell, and Download PNG saves what you are looking at. nobody needs to touch a query string. +operator, bit width and palette are dropdowns, Jump takes an $A$, an operator and a $B$ and centres that cell, and Download PNG saves the current preset as an image (for 4x4 and 8x8, that is). -the link under each image opens that exact view. - -[cb341.dev/logic-visualizer/table](https://cb341.dev/logic-visualizer/table) is the contact sheet: every operator at 8 bits, in all four palettes, side by side. it is the fastest way to see how much the palette decides what you notice. +[cb341.dev/logic-visualizer/table](https://cb341.dev/logic-visualizer/table) shows an overview: every operator at 8 bits, in all four palettes, side by side. the operators, in order: -| index | `op` | operation | -| --- | --- | --- | -| `0` | `NAND` | not (A and B) | -| `1` | `AND` | A and B | -| `2` | `NOR` | not (A or B) | -| `3` | `OR` | A or B | -| `4` | `XOR` | A xor B | -| `5`, `6` | `ADD` | addition. indices 5 and 6 label it signed and unsigned, and produce identical bits | -| `7` | `MUL` | multiplication, low bits | -| `8` | `DIV` | truncating division, B of 0 gives 0 | -| `9` | `LSHFT` | A shifted left by the low bits of B | -| `10` | `RSHFT` | A shifted right by the low bits of B | -| `11` | `CMP` | equal, less, greater as three bits | +| operation | + --- | +| `NAND` | not (A and B) | +| `AND` | A and B | +| `NOR` | not (A or B) | +| `OR` | A or B | +| `XOR` | A xor B | +| `ADD` | addition, signed and unsigned yield same result | +| `MUL` | multiplication, low bits | +| `DIV` | truncating division, B of 0 gives 0 | +| `LSHFT` | A shifted left by the low bits of B | +| `RSHFT` | A shifted right by the low bits of B | +| `CMP` | equal, less, greater as three bits | the palettes:
-how the viewer got built, ascii to shaders +so the method reaches gates, the shifter proves that much. for division it stops short and i still have to build the thing myself. -the path here was not direct. +
+how the viewer got built, ASCII to shaders -it started with the ascii table above, but printed at 8 bits, where it runs 256 rows wide. reading 1s and 0s i saw nothing, and blanking the zeros is what first made the triangle pop. +it started with the ASCII table above, but printed at 8 bits, where it runs 256 rows wide. reading 1s and 0s i saw nothing, and blanking the zeros is what first made the triangle pop. -then images. first pure black and white, one pixel per *bit*, 1 white and 0 black. neat, not informative. +then images. black and white, one pixel per *bit*, 1 white and 0 black. neat, not informative. -the first real jump was one pixel per **byte** instead of per bit. that is what makes a 256x256 image cover an entire 8-bit operator, and patterns appeared immediately. grayscale by value helped further, and the per-bit palettes came later. +what changed it was one pixel per byte instead of per bit. that is what makes a 256x256 image cover an entire 8-bit operator, and patterns showed up immediately. grayscale by value helped, and the per-bit palettes came later. but the ALU is 16-bit, and 8 bits was not the target: @@ -187,17 +188,17 @@ but the ALU is 16-bit, and 8 bits was not the target: precomputing full images was already too slow well before that. -the fix was to stop materialising the image at all: render a viewport and compute only the pixels currently on screen. that pushed the work to a fragment shader evaluating the operator per pixel on the GPU, which is also what makes panning and zooming feel like an image rather than a render queue. +the fix was to stop materialising the image at all. render a viewport, compute only the pixels currently on screen. that pushed the work to a fragment shader evaluating the operator per pixel on the GPU, which is also why panning and zooming feel like an image rather than a render queue. i used Codex for the viewer implementation, the shader, the viewport, and the tooltip plumbing. the analysis and the claims in this post are mine, and each numeric claim was checked against a brute-force pass over all 65,536 pairs rather than taken from the picture. -Tooltip showing binary, decimal, signed decimal and hex for a cell +Tooltip showing binary, decimal, signed decimal and hex for a cell *the tooltip, reading out one cell.* -it shows binary, decimal, signed decimal and hex, with the per-bit colour strips that explain what you are looking at. it is what makes the images checkable instead of decorative, since you can point at any pixel and see the arithmetic. +it shows binary, decimal, signed decimal and hex, with the per-bit colour strips. point at any pixel and you get the arithmetic for that cell, which is how i checked the readings above. -the 16-bit view still renders, though it shows the same patterns at higher resolution rather than new ones. the structure was already fully present at 8 bits, which in hindsight is the point: these are properties of the operators, not of the width. +the 16-bit view still renders and shows the same patterns at higher resolution, no new ones. the structure was already there at 8 bits. these are properties of the operators and not of the width.
@@ -205,21 +206,32 @@ the 16-bit view still renders, though it shows the same patterns at higher resol the viewer is live at [cb341.dev/logic-visualizer](https://cb341.dev/logic-visualizer/). drag to pan, wheel to zoom, arrow keys change operator, click any pixel for the readout. -![The viewer toolbar: image size, palette, zoom, jump-to-cell and download controls](/assets/blog/shape_of_logic_toolbar.png) +The viewer toolbar: image size, palette, zoom, jump-to-cell and download controls *everything is in the toolbar.* operator, bit width and palette are dropdowns, Jump takes an $A$, an operator and a $B$ and centres that cell, and Download PNG saves what you are looking at. nobody needs to touch a query string. -the links under each image above are just deep links to a view you could equally reach with the controls. if you do want to build one by hand: +the link under each image opens that exact view. -``` -?theme=0&image=8&op=MUL&a=0&b=0 -``` - -`op` takes a name or an index: `NAND`, `AND`, `NOR`, `OR`, `XOR`, `ADD`, `MUL`, `DIV`, `LSHFT`, `RSHFT`, `CMP`. +the operators, in order: -`theme` selects the palette: +| index | `op` | operation | +| --- | --- | --- | +| `0` | `NAND` | not (A and B) | +| `1` | `AND` | A and B | +| `2` | `NOR` | not (A or B) | +| `3` | `OR` | A or B | +| `4` | `XOR` | A xor B | +| `5` | `ADDA` | signed addition, same bits as `ADD` | +| `6` | `ADD` | unsigned addition | +| `7` | `MUL` | multiplication, low bits | +| `8` | `DIV` | truncating division, B of 0 gives 0 | +| `9` | `LSHFT` | A shifted left by the low bits of B | +| `10` | `RSHFT` | A shifted right by the low bits of B | +| `11` | `CMP` | equal, less, greater as three bits | + +the palettes: | value | palette | what it does | | --- | --- | --- | @@ -232,20 +244,8 @@ the links under each image above are just deep links to a view you could equally ## What's left -i still owe the divider circuit, which means sequential logic and more patience than pattern-matching asked for. - -the pictures were worth it on their own. an operator has a shape, the shape is specific to what the operation does, and you can read hardware off it: carry breaks the tiles, the shifter is 3 stages of multiplexers, the byte wraps 255 times and each wrap is a line you can point at. - -it is very obviously not random. - -## Deep links - -[^mul-link]: - -[^nand-link]: - -[^add-link]: +i still owe the divider circuit. sequential logic, and more patience than pattern-matching asked for. -[^lshft-link]: +what the pictures gave me is that an operator has a shape specific to what it does, and some of the hardware is readable from it. carry breaks the tiles. the shifter is 3 stages of multiplexers. the byte wraps 255 times and each wrap is a line you can point at. -[^div-link]: +next is building the divider in the emulator. From 6d3f7982a9348d28c9902329afff5479d31cbf98 Mon Sep 17 00:00:00 2001 From: cb341 Date: Sun, 6 Sep 2026 16:59:22 +0200 Subject: [PATCH 05/14] Update description --- _posts/2026-09-06-shape-of-logic.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/_posts/2026-09-06-shape-of-logic.md b/_posts/2026-09-06-shape-of-logic.md index 3ab3e81..80ed859 100644 --- a/_posts/2026-09-06-shape-of-logic.md +++ b/_posts/2026-09-06-shape-of-logic.md @@ -1,6 +1,6 @@ --- title: "Shape of Logic" -description: "Every input pair of an 8-bit operator fits in one 256x256 image. Bitwise ops tile by scale, addition tiles by translation, and overflow turns out to be 255 evenly spaced lines." +description: "Notation shows you one row of a truth table. A picture shows you all 65,536." tags: ["logic", "visualization", "hardware", "first principles"] math: true --- From 43da0541d02db9b5015eb4171a3b402eef147f4b Mon Sep 17 00:00:00 2001 From: cb341 Date: Sun, 6 Sep 2026 17:15:47 +0200 Subject: [PATCH 06/14] Improvements --- _posts/2026-09-06-shape-of-logic.md | 26 ++++++++++++++---- assets/blog/shape_of_logic_mul_nibble_8x8.png | Bin 0 -> 33922 bytes 2 files changed, 21 insertions(+), 5 deletions(-) create mode 100644 assets/blog/shape_of_logic_mul_nibble_8x8.png diff --git a/_posts/2026-09-06-shape-of-logic.md b/_posts/2026-09-06-shape-of-logic.md index 80ed859..2418991 100644 --- a/_posts/2026-09-06-shape-of-logic.md +++ b/_posts/2026-09-06-shape-of-logic.md @@ -1,7 +1,7 @@ --- title: "Shape of Logic" description: "Notation shows you one row of a truth table. A picture shows you all 65,536." -tags: ["logic", "visualization", "hardware", "first principles"] +tags: ["logic", "visualization", "hardware", "art"] math: true --- @@ -106,7 +106,9 @@ you can make a seam appear by picking the MSB-invert palette, which flips the co a colour map can invent structure that the data does not have. the MSB seam above is exactly that, a boundary that exists because the palette draws one. -the tiling in NAND and the bands in ADD show up under all four palettes, monochrome included, and i checked both by brute force over all pairs instead of trusting the picture. the colour makes them easier to see and does not put them there. +it cuts the other way too. nibble RGB puts 256 result values onto 121 colours, and that loss is what makes the spirals in multiplication visible at all. vibrant gives every value a distinct colour and buries the shape in detail. + +so the palette decides which scale you can see. what it cannot do is put structure into data that has none. the tiling in NAND and the bands in ADD show up under all four palettes, monochrome included, and i checked both by brute force over all pairs instead of trusting the picture. ## Shifts, and three wires @@ -128,7 +130,7 @@ back to the opening image. multiplication is the busiest operator here, and ever - powers of two are clean lines. multiplying by $2^k$ is a shift, so those rows and columns look like the shift image. - zero cells are sparse, 1,280 of 65,536, about 2%. these are the pairs whose product is divisible by 256. -- exactly 25% of results are odd. a product is odd only when both operands are odd, which is a quarter of the grid. +- a quarter of all results are odd, since a product is odd only when both operands are odd. this one is real but invisible, because the palette gives the lowest bit the same weight everywhere and nothing in the picture separates odd from even. underneath all of it is one rule. bit $k$ of $A \times B$ depends only on the low $k+1$ bits of both operands, which i checked for every bit and every pair. carry moves information upward and never downward, so the low bits of a product never learn about the high bits of its inputs. @@ -136,9 +138,23 @@ the layout follows from that. the low nibble of the product is fixed by the low the curves inside each cell are level sets of $A \times B$, which are hyperbolas. before the modulo they would be single arcs sweeping from axis to axis. the wrap at 256 chops each one into pieces, and the pieces stack into the nested rings that read as fish scales. -their spacing follows from the gradient. $A \times B$ changes at rate $\sqrt{A^2 + B^2}$, so arcs sit roughly $256 / \sqrt{A^2 + B^2}$ pixels apart. near the origin that is 22 pixels and the arcs are lazy and wide. at $A = B = 128$ it is 1.4 pixels. past $A = B = 181$ it drops below one pixel, the arcs stop being resolvable, and the corners taper into flat interference where the true product wraps 254 times before it lands. +the arcs are wide near the origin and get tighter the further out you go. small numbers multiply slowly, so you can walk a long way before the product climbs another 256 and the colour wraps. big numbers multiply fast, so the wraps come one after another. + +past the middle of the image the arcs are packed closer than one pixel apart and stop being drawable at all. that is the tapering in the corners. at $A = B = 255$ the true product is 65,025, which wraps 254 times before it lands, and there is nowhere near enough room to draw 254 separate arcs. + +the vortex look comes from arcs at different spacings crossing each other. it is a moiré pattern, the same effect you get photographing a striped shirt, and it appears because the curves are finer than the pixels available to draw them. + +the same operator in nibble RGB makes that easier to see. + +256 by 256 multiplication truth table in nibble RGB + +*multiplication, nibble RGB. [open in the viewer](https://cb341.dev/logic-visualizer/?theme=3&image=8&op=MUL)* + +four spirals, one per corner, with a cross through the middle where the arcs run out of room to curve. + +the difference is the palette. at 8 bits nibble RGB has no green, so the low nibble drives red and the high nibble drives blue, and each channel adds up the bits that are set. different results land on the same colour, 121 of them for 256 values. vibrant gives every value its own. the merging drops the fine arc-to-arc detail and leaves the slow structure, which is why the spirals come out of the noise. -the vortex look comes from that spacing gradient bending families of arcs against each other. it is a moiré pattern, an artifact of sampling curves too finely for the pixel grid, though the arithmetic underneath really is moving that fast. +neither picture is more correct. vibrant resolves individual results and buries the shape in texture, nibble RGB blurs results and shows the shape. the operator 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Date: Sun, 6 Sep 2026 17:22:36 +0200 Subject: [PATCH 07/14] Rephrase --- _posts/2026-09-06-shape-of-logic.md | 8 ++------ 1 file changed, 2 insertions(+), 6 deletions(-) diff --git a/_posts/2026-09-06-shape-of-logic.md b/_posts/2026-09-06-shape-of-logic.md index 2418991..0d8e6e4 100644 --- a/_posts/2026-09-06-shape-of-logic.md +++ b/_posts/2026-09-06-shape-of-logic.md @@ -136,13 +136,9 @@ underneath all of it is one rule. bit $k$ of $A \times B$ depends only on the lo the layout follows from that. the low nibble of the product is fixed by the low nibbles of the inputs alone, which draws the 16x16 grid of cells repeating their fine detail. the high bits depend on everything, so they vary slowly and paint the 4x4 arrangement of large squares on top. the nibble grid you can see is that split between fast and slow bits. -the curves inside each cell are level sets of $A \times B$, which are hyperbolas. before the modulo they would be single arcs sweeping from axis to axis. the wrap at 256 chops each one into pieces, and the pieces stack into the nested rings that read as fish scales. +the curves inside each cell are the pairs that share a product. the wrap at 256 cuts them into the nested rings that read as fish scales. -the arcs are wide near the origin and get tighter the further out you go. small numbers multiply slowly, so you can walk a long way before the product climbs another 256 and the colour wraps. big numbers multiply fast, so the wraps come one after another. - -past the middle of the image the arcs are packed closer than one pixel apart and stop being drawable at all. that is the tapering in the corners. at $A = B = 255$ the true product is 65,025, which wraps 254 times before it lands, and there is nowhere near enough room to draw 254 separate arcs. - -the vortex look comes from arcs at different spacings crossing each other. it is a moiré pattern, the same effect you get photographing a striped shirt, and it appears because the curves are finer than the pixels available to draw them. +they are wide near the origin, where products grow slowly, and tighten as the numbers get bigger. in the corners they are too fine to draw and blur into the tapering, and where sets of them cross at different spacings you get the vortices. the same operator in nibble RGB makes that easier to see. From b703f88d0d64f51a8465fe1662faf067da0e6f78 Mon Sep 17 00:00:00 2001 From: cb341 Date: Sun, 6 Sep 2026 17:25:38 +0200 Subject: [PATCH 08/14] Reduce asset size --- _posts/2026-09-06-shape-of-logic.md | 6 +++--- assets/blog/shape_of_logic_tc_major.png | Bin 554111 -> 0 bytes assets/blog/shape_of_logic_tc_major.webp | Bin 0 -> 44196 bytes assets/blog/shape_of_logic_toolbar.png | Bin 107113 -> 0 bytes assets/blog/shape_of_logic_toolbar.webp | Bin 0 -> 48664 bytes assets/blog/shape_of_logic_tooltip.png | Bin 60904 -> 0 bytes assets/blog/shape_of_logic_tooltip.webp | Bin 0 -> 25696 bytes 7 files changed, 3 insertions(+), 3 deletions(-) delete mode 100644 assets/blog/shape_of_logic_tc_major.png create mode 100644 assets/blog/shape_of_logic_tc_major.webp delete mode 100644 assets/blog/shape_of_logic_toolbar.png create mode 100644 assets/blog/shape_of_logic_toolbar.webp delete mode 100644 assets/blog/shape_of_logic_tooltip.png create mode 100644 assets/blog/shape_of_logic_tooltip.webp diff --git a/_posts/2026-09-06-shape-of-logic.md b/_posts/2026-09-06-shape-of-logic.md index 0d8e6e4..c2a2606 100644 --- a/_posts/2026-09-06-shape-of-logic.md +++ b/_posts/2026-09-06-shape-of-logic.md @@ -174,7 +174,7 @@ this started in [turing complete 2.0](https://store.steampowered.com/app/1444480 i wanted to write it up in late july. the time only showed up now, which meant coming back to the images cold and checking the claims instead of trusting what i remembered seeing. -Turing Complete major update announcement +Turing Complete major update announcement the ALU had one-cycle addition, subtraction, and multiplication. i wanted division on the same terms, a dedicated circuit that answers in one pass instead of a loop that shifts and subtracts once per bit. the usual restoring divider costs a cycle per output bit, which for 16 bits is 16 cycles against multiplication's one. @@ -204,7 +204,7 @@ the fix was to stop materialising the image at all. render a viewport, compute o i used Codex for the viewer implementation, the shader, the viewport, and the tooltip plumbing. the analysis and the claims in this post are mine, and each numeric claim was checked against a brute-force pass over all 65,536 pairs rather than taken from the picture. -Tooltip showing binary, decimal, signed decimal and hex for a cell +Tooltip showing binary, decimal, signed decimal and hex for a cell *the tooltip, reading out one cell.* @@ -218,7 +218,7 @@ the 16-bit view still renders and shows the same patterns at higher resolution, the viewer is live at [cb341.dev/logic-visualizer](https://cb341.dev/logic-visualizer/). drag to pan, wheel to zoom, arrow keys change operator, click any pixel for the readout. -The viewer toolbar: image size, palette, zoom, jump-to-cell and download controls +The viewer toolbar: image size, palette, zoom, jump-to-cell and download controls *everything is in the toolbar.* diff --git a/assets/blog/shape_of_logic_tc_major.png b/assets/blog/shape_of_logic_tc_major.png 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zecunqvL?MfWVsxwRXtD7SPkQ1a>?;cTP@{B6EHWI zVxOU10Vy5Bl z8yoJ)@gBjV4g9IWnrR~*0)22c{aIU!Fm&)^Y}SWDXYuoR#S;roJY@#~wrs-bO^tX9 ztNs+Po)m*nWJXB67AmoJS4#-edO7`o?%qP2VGsk;<=J(ijW5)~j^I~R0143R^pVzP z3;+ND+M3~@00e3eQ~&_I-^nWE@pRNE69ry$kN}&&U9}sbfB+mq00Dl|;0jx7c_hi2 zebc36v2&dnA(7VG2O4fO&xC0HlUxveKfFo6yz1qgUr!^j3t)t|AocVyzRxcfUO&w% z;|sVoK5hZ;dYK-L0i7oFx!^dUvC<{8G@BUM6&pbl4+zhKrhTRlXh2eicM|%(`8hhD zKy#sO6Qk2X&0?_LzBp1od#S2uK8_tKZc<2QbSLb)wPoy~QjA73J8oO6Yjm(!o#?_6 zPDLGBG!h!DLwE$-Q($wX1LQgf6oa~ALf3lfyQPRj!bnpvnU6$K)Vq{!G6P|jD$?Zc zry&=m$z0UDozA}Bz Date: Sun, 6 Sep 2026 17:46:06 +0200 Subject: [PATCH 09/14] Rewrite + mention 16x16 --- _posts/2026-09-06-shape-of-logic.md | 18 +++++++++++++++--- assets/blog/shape_of_logic_mystery_16.webp | Bin 0 -> 120786 bytes 2 files changed, 15 insertions(+), 3 deletions(-) create mode 100644 assets/blog/shape_of_logic_mystery_16.webp diff --git a/_posts/2026-09-06-shape-of-logic.md b/_posts/2026-09-06-shape-of-logic.md index c2a2606..0badb66 100644 --- a/_posts/2026-09-06-shape-of-logic.md +++ b/_posts/2026-09-06-shape-of-logic.md @@ -9,7 +9,7 @@ an 8-bit operator takes two bytes and returns one. its truth table has $256 \tim 256 by 256 multiplication truth table -*multiplication, every input pair at once. [open in the viewer](https://cb341.dev/logic-visualizer/?theme=0&image=8&op=MUL)* +*multiplication, every input pair at once. [open in the viewer](https://cb341.dev/logic-visualizer/?theme=2&image=8&op=MUL)* ## The idea @@ -64,7 +64,7 @@ for all 65,536 pairs, the low nibble of $A \mathbin{\overline{\wedge}} B$ depend addition truth table in nibble RGB -*addition at 16 bits, nibble RGB. blocks become bands. [open in the viewer](https://cb341.dev/logic-visualizer/?theme=3&image=16&op=ADD)* +*addition at 8 bits, nibble RGB. blocks become bands. [open in the viewer](https://cb341.dev/logic-visualizer/?theme=2&image=8&op=ADD)* carry. bit $k$ of $A + B$ depends on every bit below $k$, so the bits are no longer independent and the nested blocks go with them. the nibble test that passes for NAND fails here, because the low nibble of a sum can be changed by a carry out of it. @@ -210,10 +210,22 @@ i used Codex for the viewer implementation, the shader, the viewport, and the to it shows binary, decimal, signed decimal and hex, with the per-bit colour strips. point at any pixel and you get the arithmetic for that cell, which is how i checked the readings above. -the 16-bit view still renders and shows the same patterns at higher resolution, no new ones. the structure was already there at 8 bits. these are properties of the operators and not of the width. +the 16-bit view renders the same way, and the patterns hold up rather than dissolving. the operators keep their character at both widths.