# 2025
video (5MB)
A visual proof that
(1+2+3+4+5+6+7+8+9)² = 1³+2³+3³+4³+5³+6³+7³+8³+9³ = 2025
# 1 Source Code
video.c:
#include <math.h>
#include <stdio.h>
#include <stdlib.h>
#include <string.h>
#ifndef M_PI
#define M_PI 3.14159265358979
#endif
float anchor[11][2025][4];
float position[2025][4];
float velocity[2025][4];
#define H 1080
#define W 1920
#define r 2
#define aa 4
unsigned char ppm[H * aa][W * aa][3];
int main(int argc, char **argv)
{
(void) argc;
(void) argv;
memset(anchor, 0, sizeof(anchor));
float *p = &anchor[0][0][0];
for (int j = 1; j <= 9; ++j)
{
for (int jj = 1; jj <= j; ++jj)
{
for (int i = 1; i <= 9; ++i)
{
for (int ii = 1; ii <= i; ++ii)
{
int x = - (j - 1) * 10 - (4 - 1) + (4 - 1) * 2 + (jj - 1) * 20;
int y = 12 + (j - 1) * 20 + (4 - 1) * 2;
*p++ = x / 179.0 * 2;
*p++ = y * sqrt(3)/2 / 179.0 * 2 - 1;
*p++ = 1;
*p++ = 1;
}
}
}
}
for (int j = 1; j <= 9; ++j)
{
for (int jj = 1; jj <= j; ++jj)
{
for (int i = 1; i <= 9; ++i)
{
for (int ii = 1; ii <= i; ++ii)
{
int x = - (j - 1) * 10 - (i - 1) + (ii - 1) * 2 + (jj - 1) * 20;
int y = 12 + (j - 1) * 20 + (i - 1) * 2;
*p++ = x / 179.0 * 2;
*p++ = y * sqrt(3)/2 / 179.0 * 2 - 1;
*p++ = 1;
*p++ = 1;
}
}
}
}
#if 0
for (int j = 0; j < 45; ++j)
{
for (int i = 0; i < 45; ++i)
{
*p++ = (i / 44.0 * 2 - 1) * 0.8;
*p++ = (j / 44.0 * 2 - 1) * 0.8;
*p++ = 0;
*p++ = 1;
}
}
#endif
for (int j = 1; j <= 9; ++j)
{
for (int i = 1; i <= 9; ++i)
{
for (int jj = 1; jj <= j; ++jj)
{
for (int ii = 1; ii <= i; ++ii)
{
int x = (i - 1) + (ii - 1) + i * (i - 1) / 2;
int y = (j - 1) + (jj - 1) + j * (j - 1) / 2;
*p++ = (x / 52.0 * 2 - 1) * 0.8;
*p++ = (y / 52.0 * 2 - 1) * 0.8;
*p++ = 1;
*p++ = 1;
}
}
}
}
#define PERMUTED(inner) \
for (int k = 1; k <= 9; ++k) \
{ \
for (int l = 1; l <= k; ++l) \
{ \
for (int i = 1; i <= k; ++i) \
{ \
for (int ii = 1; ii <= i; ++ii) \
{ \
int j = k; \
int jj = l; \
inner \
} \
} \
for (int j = k - 1; j >= 1; --j) \
{ \
for (int jj = 1; jj <= j; ++jj) \
{ \
int i = k; \
int ii = l; \
inner \
} \
} \
} \
} \
#define inner \
int x = (i - 1) + (ii - 1) + i * (i - 1) / 2; \
int y = (j - 1) + (jj - 1) + j * (j - 1) / 2; \
*p++ = (x / 52.0 * 2 - 1) * 0.8; \
*p++ = (y / 52.0 * 2 - 1) * 0.8; \
*p++ = 1; \
*p++ = 1;
PERMUTED(inner)
#undef inner
for (int k = 0; k < 2025; ++k)
{
position[k][0] = 0*anchor[0][k][0] + 0.01 * (rand() / (double) RAND_MAX * 2 - 1);
position[k][1] = 0*anchor[0][k][1] + 0.01 * (rand() / (double) RAND_MAX * 2 - 1);
position[k][2] = 0*anchor[0][k][2] + 0.01 * (rand() / (double) RAND_MAX * 2 - 1);
}
const float dt = 1.0/60.0;
for (int frame = 0; frame < 960; ++frame)
{
memset(ppm, 0, sizeof(ppm));
float t = frame / (10.0 * 60.0) * 4;
t *= 2;
if (t > 4) t = (t - 4) / 2 + 4;
if (t > 6) t = (t - 6) * 3 + 6;
if (t > 8) t = (t - 8) / 3 * 2 + 8;
int w = floor(t);
if (w > 10) w = 10;;
for (int k = 0; k < 2025; ++k)
{
switch (w)
{
case 0:
case 1:
case 2:
case 3:
break;
case 4:
{
float s = 2 * M_PI * (t - 4) / 4;
float co = cosf(s);
float si = sinf(s);
float *p = &anchor[w][0][0];
#define inner \
float x = (i - 1) + (ii - 1) + i * (i - 1) / 2; \
float y = (j - 1) + (jj - 1) + j * (j - 1) / 2; \
if (i > j) \
{ \
float x0 = (i - 1) + (0.5 * (i + 1) - 1) + i * (i - 1) / 2; \
float y0 = x0; \
x -= x0; \
y -= y0; \
float x1 = co * x - si * y; \
float y1 = si * x + co * y; \
x = x1 + x0; \
y = y1 + y0; \
} \
*p++ = (x / 52.0 * 2 - 1) * 0.8; \
*p++ = (y / 52.0 * 2 - 1) * 0.8; \
*p++ = 1; \
*p++ = 1;
PERMUTED(inner)
#undef inner
break;
}
case 5:
{
float s = 2 * M_PI * (t - 5) / 4;
float co = cosf(s);
float si = sinf(s);
float *p = &anchor[w][0][0];
#define inner \
float x = (i - 1) + (ii - 1) + i * (i - 1) / 2; \
float y = (j - 1) + (jj - 1) + j * (j - 1) / 2; \
float z = 0; \
if (i > j) \
{ \
float x0 = (i - 1) + (0.5 * (i + 1) - 1) + i * (i - 1) / 2; \
float y0 = x0; \
x -= x0; \
y -= y0; \
float x1 = - y; \
float y1 = x; \
x = x1 + x0; \
y = y1 + y0; \
z = si * (y - y0); \
y = co * (y - y0) + y0; \
} \
else \
{ \
float x0 = (j - 1) + (0.5 * (j + 1) - 1) + j * (j - 1) / 2; \
float y0 = x0; \
z = si * (y - y0); \
y = co * (y - y0) + y0; \
} \
*p++ = (x / 52.0 * 2 - 1) * 0.8; \
*p++ = (y / 52.0 * 2 - 1) * 0.8; \
*p++ = (z / 52.0 * 2 ) * 0.8 + 1; \
*p++ = 1;
PERMUTED(inner)
#undef inner
break;
}
case 6:
{
float *p = &anchor[w][0][0];
#define inner \
float x = (i - 1) + (ii - 1) + i * (i - 1) / 2; \
float y = (j - 1) + (jj - 1) + j * (j - 1) / 2; \
float z = 0; \
if (i > j) \
{ \
float x0 = (i - 1) + (0.5 * (i + 1) - 1) + i * (i - 1) / 2; \
float y0 = x0; \
x -= x0; \
y -= y0; \
float x1 = - y; \
float y1 = x; \
x = x1 + x0; \
y = y1 + y0; \
z = y - y0; \
y = y0 + (j - i); \
} \
else \
{ \
float x0 = (j - 1) + (0.5 * (j + 1) - 1) + j * (j - 1) / 2; \
float y0 = x0; \
z = y - y0; \
y = y0 + (j - i); \
} \
*p++ = (x / 52.0 * 2 - 1) * 0.8; \
*p++ = (y / 52.0 * 2 - 1) * 0.8; \
*p++ = (z / 52.0 * 2 ) * 0.8 + 1; \
*p++ = 1;
PERMUTED(inner)
#undef inner
break;
}
case 7:
{
float *p = &anchor[w][0][0];
#define inner \
float x = (i - 1) + (ii - 1) + i * (i - 1) / 2; \
float y = (j - 1) + (jj - 1) + j * (j - 1) / 2; \
float z = 0; \
if (i > j) \
{ \
float x0 = (i - 1) + (0.5 * (i + 1) - 1) + i * (i - 1) / 2; \
float y0 = x0; \
x -= x0; \
y -= y0; \
float y1 = x; \
x = x0 - jj + (0.5 * (i + 1)); \
y = y1 + y0; \
z = y - y0; \
y = y0 + (j - i); \
} \
else \
{ \
float x0 = (j - 1) + (0.5 * (j + 1) - 1) + j * (j - 1) / 2; \
float y0 = x0; \
z = y - y0; \
y = y0 + (j - i); \
x = x0 + ii - (0.5 * (j + 1)); \
} \
*p++ = (x / 52.0 * 2 - 1) * 0.8; \
*p++ = (y / 52.0 * 2 - 1) * 0.8; \
*p++ = (z / 52.0 * 2 ) * 0.8 + 1; \
*p++ = 1;
PERMUTED(inner)
#undef inner
break;
}
case 8:
{
float *p = &anchor[w][0][0];
#define inner \
float x = (i - 1) + (ii - 1) + i * (i - 1) / 2; \
float y = (j - 1) + (jj - 1) + j * (j - 1) / 2; \
float z = 0; \
if (i > j) \
{ \
float x0 = (i - 1) + (0.5 * (i + 1) - 1) + i * (i - 1) / 2; \
float y0 = x0; \
x -= x0; \
y -= y0; \
float y1 = x; \
x = x0 - jj + (0.5 * (i + 1)); \
y = y1 + y0; \
z = y - y0; \
y = y0 + (j - i) + (i - 1 - 2 * (jj - 1)) * 0.5; \
} \
else \
{ \
float x0 = (j - 1) + (0.5 * (j + 1) - 1) + j * (j - 1) / 2; \
float y0 = x0; \
z = y - y0; \
y = y0 + (j - i) - (j - 1 - 2 * (ii - 1)) * 0.5; \
x = x0 + ii - (0.5 * (j + 1)); \
} \
*p++ = (x / 52.0 * 2 - 1) * 0.8; \
*p++ = (y / 52.0 * 2 - 1) * 0.8; \
*p++ = (z / 52.0 * 2 ) * 0.8 + 1; \
*p++ = 1;
PERMUTED(inner)
#undef inner
break;
}
case 9:
{
float *p = &anchor[w][0][0];
#define inner \
float x00 = (((k - 1) + k * (k - 1) / 2) - 52.0 / 2.0) * 1.5 + 52.0 / 2.0; \
float y00 = 52.0 / 2.0; \
float x = (i - 1) + (ii - 1) + i * (i - 1) / 2; \
float y = (j - 1) + (jj - 1) + j * (j - 1) / 2; \
float z = 0; \
if (i > j) \
{ \
float x0 = (i - 1) + (0.5 * (i + 1) - 1) + i * (i - 1) / 2; \
float y0 = x0; \
x00 -= x0; \
y00 -= y0; \
x -= x0; \
y -= y0; \
float y1 = x; \
x = x0 - jj + (0.5 * (i + 1)); \
y = y1 + y0; \
z = y - y0; \
y = y0 + (j - i) + (i - 1 - 2 * (jj - 1)) * 0.5; \
} \
else \
{ \
float x0 = (j - 1) + (0.5 * (j + 1) - 1) + j * (j - 1) / 2; \
float y0 = x0; \
x00 -= x0; \
y00 -= y0; \
z = y - y0; \
y = y0 + (j - i) - (j - 1 - 2 * (ii - 1)) * 0.5; \
x = x0 + ii - (0.5 * (j + 1)); \
} \
x += x00; \
y += y00; \
*p++ = (x / 52.0 * 2 - 1) * 0.8; \
*p++ = (y / 52.0 * 2 - 1) * 0.8; \
*p++ = (z / 52.0 * 2 ) * 0.8 + 1; \
*p++ = 1;
PERMUTED(inner)
#undef inner
break;
}
case 10:
{
float *p = &anchor[w][0][0];
for (int k = 0; k < 2025; ++k)
{
*p++ = 0;
*p++ = 0;
*p++ = 1;
*p++ = 1;
}
break;
}
default:
return 0;
}
float cx = t < 4 ? 0 : t < 5 ? t - 4 : t < 7 ? 1 : 0;
float cy = t < 6 ? 0 : t < 9 ? 1 : 0;
float t = 0.3 - 2 * M_PI * k / 2025;
unsigned char rgb[3] = { 128 + 127 * cosf(t), 128 + 127 * cosf(t + 2), 128 + 127 * cosf(t + 4) };
float dx = position[k][0] - (anchor[w][k][0] - 0.6 * (1 - cos(M_PI * cx)) / 2);
float dy = position[k][1] - (anchor[w][k][1] - 0.1 * (1 - cos(M_PI * cy)) / 2);
float dz = position[k][2] - anchor[w][k][2];
float d2 = dx * dx + dy * dy + dz * dz;
float d = sqrtf(d2); if (! (d > 0)) d = 1;
float F = 100 * d;
float f = 0.8;
velocity[k][0] *= f;
velocity[k][1] *= f;
velocity[k][2] *= f;
velocity[k][0] -= F * dx / d * dt;
velocity[k][1] -= F * dy / d * dt;
velocity[k][2] -= F * dz / d * dt;
position[k][0] += velocity[k][0] * dt;
position[k][1] += velocity[k][1] * dt;
position[k][2] += velocity[k][2] * dt;
int x0 = (position[k][0] * (2 - position[k][2]) * 0.9 / 2 * H + W / 2) * aa;
int y0 = (position[k][1] * (2 - position[k][2]) * 0.9 / 2 * H + H / 2) * aa;
float r0 = r * (2 - position[k][2]);
for (int j = -floor(r0 * aa); j <= ceil(r0 * aa); ++j)
{
int y = y0 + j;
for (int i = -floor(r0 * aa); i <= ceil(r0 * aa); ++i)
{
int x = x0 + i;
if (0 <= x && x < W * aa && 0 <= y && y < H * aa && i * i + j * j <= r0 * aa * r0 * aa)
{
ppm[y][x][0] = rgb[0];
ppm[y][x][1] = rgb[1];
ppm[y][x][2] = rgb[2];
}
}
}
}
printf("P6\n%d %d\n255\n", W * aa, H * aa);
fwrite(ppm, sizeof(ppm), 1, stdout);
}
return 0;
}
Makefile:
C99 = gcc -std=c99 -Wall -Wextra -pedantic -O3
all: video.mp4
%.exe: %.c
$(C99) -o $@ $< -lm
%.mp4: %.exe
for pass in 1 2 3 ; do ./$< | ffmpeg -r 60 -framerate 60 -i - -s 1920x1080 -pix_fmt yuv420p -profile:v high -level:v 4.2 -b:v 2.5M -g:v 30 -bf:v 2 -pass $$pass -movflags +faststart -y $@ ; done