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The 34 Convex Heptahedra and their Characteristic
Polynomials
Followed by free programs you can use as you untangle
the 257 convex octahedra.
x (x2+x-1) (x3-x2-9x-4) |
x (x+1) (x+2) (x3-3x2-4x+4) |
(x-1)2 (x+1)2 (x+2) (x2-2x-6) |
(x+2) (x2-2) (x4-2x3-6x2+2x+2) |
x7-12x5-10x4+23x3+20x2-9x-6 |
x (x-1) (x+1) (x4-11x2-8x+8) |
x (x2-2)(x4-10x2-8x+2) |
x2 (x+2)2 (x3-4x2+6) |
(x2+x-1)2 (x3-2x2-7x+6) |
x (x+2) (x5-2x4-8x3+8x2+8x-6) |
(x-1) (x+1)2 (x2-3x-1) (x3+2x2-4x-7) |
(x-1) (x+1) (x+2) (x5-2x4-8x3+8x2+11x+2) |
(x4-x3-8x2+5) (x4+x3-4x2-4x+1) |
x8-13x6-8x5+42x4+34x3-35x2-26x+4 |
(x3+2x2-x-1) (x5-2x4-8x3+9x2+11x-9) |
(x3+x2-2x-1) (x5-x4-10x3+3x2+18x-4) |
(x-1)(x+2)(x2-2)(x4-x3-8x2+4) |
x8-13x6-6x5+40x4+20x3-38x2-14x+9 |
(x+1) (x7-x6-12x5+6x4+32x3-10x2-14x+4) |
(x4+x3-4x2+1) (x4-x3-8x2+5) |
(x-1) (x+1) (x2+x-1) (x4-x3-10x2+5x+9) |
(x+2) (x3-3x-1) (x5-2x4-7x3+9x2+6x-2) |
(x+1) (x3+x2-2x-1) (x5-2x4-9x3+13x2+15x-8) |
(x3+x2-2x-1) (x6-x5-11x4+4x3+32x2-x+22) |
(x3+2x2-x-1) (x6-2x5-9x4+11x3+24x2-12x-14) |
x (x-2) (x-1) (x+1)2 (x+2) (x3-x2-8x+4) |
(x2-x-1) (x2+x-1) (x2+3x+1) (x3-3x2-3x+8) |
x9-14x7-4x6+55x5+18x4-70x3-20x2+25x+8 |
(x3+x2-2x-1) (x6-x5-11x4+8x3+20x2-7x-8) |
(x-3) (x-1) (x+2)2 (x3-3x-1)2 |
(x-3) (x2-x-1) (x2+x-1) (x2+3x+1) (x3-4x-1) |
x (x-3) (x-1) (x+1) (x+2) (x5+x4-7x3-5x2+10x+4) |
(x-3) (x3+x2-2x-1)2 (x3+x2-5x-1) |
(x-3)(x-1)(x2-x-1)2(x2+3x+1)2 |
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I've done an interesting study using Matrix calculator and Factoris . There are 7 convex hexahedra, and 34 convex heptahedra. It is fairly easy to figure out the characteristic polynomial of each one when such tools are available. I've started with a somewhat irregular six-sided figure. Note that points A and C are connected. Also, note that A and F are not connected. To make an adjacency matrix, one merely makes a grid of ones and zeros. Ones indicate a connection, or an adjacency. Zeroes indicate no connection. Since the unlying graph has six points (A B C D E and F), it has a 6x6 adjacency matrix. You can copy the matrix in the third box, and paste it into the Matrix calculator . Then indicate you want the characteristic polynomial. You'll get x6 - 10x4 - 8x3 + 9x2 + 4x - 1. If you relabel the graph in a different way (perhaps A C E F B D) and create the new adjacency matrix corresponding to this new labeling, you'll still get the same polynomial.
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0 1 1 0 1 0
1 0 0 1 1 1 1 0 0 1 1 0 0 1 1 0 0 1 1 1 1 0 0 1 0 1 0 1 1 0 |
0,1,1,1,1,1,0,0,0,0,0,0
1,0,1,0,1,0,1,0,0,0,1,0
1,1,0,1,0,0,0,0,0,1,1,0
1,0,1,0,0,1,0,0,1,1,0,0
1,1,0,0,0,1,1,1,0,0,0,0
1,0,0,1,1,0,0,1,1,0,0,0
0,1,0,0,1,0,0,1,0,0,1,1
0,0,0,0,1,1,1,0,1,0,0,1
0,0,0,1,0,1,0,1,0,1,0,1
0,0,1,1,0,0,0,0,1,0,1,1
0,1,1,0,0,0,1,0,0,1,0,1
0,0,0,0,0,0,1,1,1,1,1,0
An adjacency matrix of the icosahedron
![]() -x (x + 2) (x5 - 2x4 - 12x3 - 6x2 + 10x + 6) |
![]() (x + 1)(x5 - x4 - 9x3 - 3x2 + 10x + 4) |