Hexany - Coordinates For The Pascal's Triangle of Combination Product Sets

Coordinates For The Pascal's Triangle of Combination Product Sets

First row (square):
00
10 01
11

Second row (cube or octony):
000
100 010 001 triad (triangle)
110 101 011 triad (triangle)
111

Third row (hypercube)
0000
1000 0100 0010 0001 tetrad (tetrahedron or 3-simplex)
1100 1010 1001 0110 0101 0011 hexany (octahedron)
1110 1101 1011 0111 tetrad
1111

The octahedron there is the edge dual of the tetrahedron, or rectified tetrahedron

Fourth row (5-dimensional cube)
00000
10000 01000 00100 00010 00001 pentad (4-simplex or pentachoron - four dimensional tetrahedron)
11000 10100 10010 10001 01100 01010 01001 00110 00101 00011 2)5 dekany (10 vertices, rectified 4-simplex)
00111 01011 01101 01110 10011 10101 10110 11001 11010 11100 3)5 dekany (10 vertices)
01111 10111 11011 11101 11110 pentad
11111

The rectified 4-simplex for the dekany is also known as the dispentachoron

Fifth row (6-dimensional cube
000000
100000 010000 001000 000100 000010 000001 hexad (5-simplex or hexateron - five dimensional tetrahedron)
110000 101000 100100 100010 100001 011000 010100 010010 010001 001100 001010 001001 000110 000101 000011 2)6 pentadekany (15 vertices, rectified 5-simplex)
111000 110100 110010 110001 101100 101010 101001 100110 100101 100011 011100 011010 011001 010110 010101 010011 001110 001101 001011 000111 eikosany (20 vertices birectified 5-simplex)
001111 010111 011011 011101 011110 100111 101011 101101 101110 110011 110101 110110 111001 111010 111100 4)6 pentadekany (15 vertices)
011111 101111 110111 111011 111101 111110 hexad
111111

It is easy to see that the geometric figure for the dekany is the edge dual of the 4-simplex and the one for the pentadekany is the edge dual of the 5-simplex.

To see this, in the figure of the octahedron in the hypercube, scale the entire figure by 1/2 about the origin (blue vertex). The octahedron vertices will move to the midpoints of the original tetrahedron edges (joining the red vertices in the figure).

So - similarly the dekany vertices when scaled by 1/2 move to the midpoints of the 4-simplex edges, and the pentadekany vertices move to the midpoints of the 5-simplex edges, and so on in all the higher dimensions.

The eikosany vertices when scaled by 1/3 move to the centres of the 2D faces of the 5-simplex. To see that, note that in a 3D cube, 111 when scaled by 1/3 moves to the midpoint of 100 010 001 (each edge vector subtends the same distance along the long diagonal of the cube). So 11100 moves to the centre of the equilateral triangle with coords 10000 01000 00100 and similarly for all the other eikosany vertices.

So - the geometric figure for the eikosany is the 2D face dual of the 5-simplex or birectified 5-simplex. Similarly for the 3)7, 3)8 etc. figures in all higher dimensions.

Similarly in eight dimensions, the figure you get using all permutations of 4 out of 8 is the 3D face dual of the 7-simplex, or 3-rectified 7-simplex (since 1111 scaled by 1/4 moves to the centre of the 3D regular tetrahedron face 1000 0100 0010 0001), and so on.