This image will surely be familiar to many people on this forum:
CMT = RGBColor /@ Drop[Import["http://inversed.ru/Blog/Colormaps/New_Laguna.csv"], 4];
CM[x_] := Blend[CMT, x];
n = 8;
M = 2^n;
ArrayPlot[
Table[BitXor[i, j], {i, 0, M - 1}, {j, 0, M - 1}],
ColorFunction -> CM,
PixelConstrained -> 4,
Frame -> None
]
Its values are calculated by taking the bitwise xor of x and y coordinates, and its level sets (x xor y > t) form the famous munching squares animation discovered back in 1962:
A more interesting image is obtained by plotting the bit counts of x xor y:
n = 8;
M = 2^n;
ArrayPlot[
Table[
Total[IntegerDigits[BitXor[i, j], 2, n]],
{i, 0, M - 1}, {j, 0, M - 1}
],
ColorFunction -> CM,
PixelConstrained -> 4,
Frame -> None
]
Let’s see what happens if we convert the coordinates to Gray codes before the xor (BinToGray[x_] := BitXor[x, Quotient[x, 2]]). The function itself:
BinToGray[x_] := BitXor[x, Quotient[x, 2]];
GrayToBin[x_] := Module[
{z = x, y = Quotient[x, 2]},
While[y != 0,
z = BitXor[z, y];
y = Quotient[y, 2]
];
z
];
n = 9;
M = 2^n;
ToDigits[x_] := IntegerDigits[BinToGray[x], 2, n];
ArrayPlot[
Table[
FromDigits[Mod[ToDigits[i] + ToDigits[j], 2], 2],
{i, 0, M - 1}, {j, 0, M - 1}
],
ColorFunction -> CM,
PixelConstrained -> 2,
Frame -> None
]
Modified munching squares:
n = 9;
M = 2^n;
ToDigits[x_] := IntegerDigits[BinToGray[x], 2, n];
A = Table[
FromDigits[Mod[ToDigits[i] + ToDigits[j], 2], 2], {i, 0,
M - 1}, {j, 0, M - 1}];
Draw[L_] := Rasterize[ArrayPlot[
Table[Boole[
\!\(\*SubscriptBox[\(A\), \(\(\[LeftDoubleBracket]\)\(i,
j\)\(\[RightDoubleBracket]\)\)]\) < L], {i, 1, M}, {j, 1,
M}],
ColorFunction -> GrayLevel,
ColorFunctionScaling -> False,
PixelConstrained -> 1,
Frame -> None
]];
Draw[311]
NFrames = M + 1;
ExportFrame[i_] :=
Export[
"C:\\Temp\\" <> IntegerString[i, 10, 4] <> ".png",
Draw[i]
];
ParallelMap[ExportFrame, Range[0, NFrames - 1]];
But bit count is the real beauty:
n = 9;
M = 2^n;
ToDigits[x_] := IntegerDigits[BinToGray[x], 2, n];
ArrayPlot[
Table[
Total[Mod[ToDigits[i] + ToDigits[j], 2]],
{i, 0, M - 1}, {j, 0, M - 1}
],
ColorFunction -> CM,
PixelConstrained -> 2,
Frame -> None
]
XorArraysForum.nb (618 KB)
