What is the maximal Euclidean distance between any two of these nine points?

Geometry Level 1

This problem’s question: {\color{#D61F06}\text{This problem's question:}} What is the maximal Euclidean distance between any two of these nine points?

The points are eight-dimensional points. The Euclidean distance is nonetheless defined.

( 2 3 0 0 0 0 0 0 0 − 1 12 7 4 0 0 0 0 0 0 − 1 12 − 1 4 7 3 7 0 0 0 0 0 − 1 12 − 1 4 7 − 1 2 21 5 3 2 0 0 0 0 − 1 12 − 1 4 7 − 1 2 21 − 1 2 15 2 5 0 0 0 − 1 12 − 1 4 7 − 1 2 21 − 1 2 15 − 1 2 10 3 2 2 0 0 − 1 12 − 1 4 7 − 1 2 21 − 1 2 15 − 1 2 10 − 1 2 6 1 3 0 − 1 12 − 1 4 7 − 1 2 21 − 1 2 15 − 1 2 10 − 1 2 6 − 1 2 3 1 2 − 1 12 − 1 4 7 − 1 2 21 − 1 2 15 − 1 2 10 − 1 2 6 − 1 2 3 − 1 2 ) \left( \begin{array}{cccccccc} \frac{2}{3} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ -\frac{1}{12} & \frac{\sqrt{7}}{4} & 0 & 0 & 0 & 0 & 0 & 0 \\ -\frac{1}{12} & -\frac{1}{4 \sqrt{7}} & \sqrt{\frac{3}{7}} & 0 & 0 & 0 & 0 & 0 \\ -\frac{1}{12} & -\frac{1}{4 \sqrt{7}} & -\frac{1}{2 \sqrt{21}} & \frac{\sqrt{\frac{5}{3}}}{2} & 0 & 0 & 0 & 0 \\ -\frac{1}{12} & -\frac{1}{4 \sqrt{7}} & -\frac{1}{2 \sqrt{21}} & -\frac{1}{2 \sqrt{15}} & \sqrt{\frac{2}{5}} & 0 & 0 & 0 \\ -\frac{1}{12} & -\frac{1}{4 \sqrt{7}} & -\frac{1}{2 \sqrt{21}} & -\frac{1}{2 \sqrt{15}} & -\frac{1}{2 \sqrt{10}} & \frac{\sqrt{\frac{3}{2}}}{2} & 0 & 0 \\ -\frac{1}{12} & -\frac{1}{4 \sqrt{7}} & -\frac{1}{2 \sqrt{21}} & -\frac{1}{2 \sqrt{15}} & -\frac{1}{2 \sqrt{10}} & -\frac{1}{2 \sqrt{6}} & \frac{1}{\sqrt{3}} & 0 \\ -\frac{1}{12} & -\frac{1}{4 \sqrt{7}} & -\frac{1}{2 \sqrt{21}} & -\frac{1}{2 \sqrt{15}} & -\frac{1}{2 \sqrt{10}} & -\frac{1}{2 \sqrt{6}} & -\frac{1}{2 \sqrt{3}} & \frac{1}{2} \\ -\frac{1}{12} & -\frac{1}{4 \sqrt{7}} & -\frac{1}{2 \sqrt{21}} & -\frac{1}{2 \sqrt{15}} & -\frac{1}{2 \sqrt{10}} & -\frac{1}{2 \sqrt{6}} & -\frac{1}{2 \sqrt{3}} & -\frac{1}{2} \\ \end{array} \right)

All 36 unique distances were computed. The distance is the same no matter which point is listed first in the computation; therefore, the number of distances to be computed is halved. If you recognize what these points are, then the problem becomes quite simple.


The answer is 1.

A Former Brilliant Member by A Former Brilliant Member

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1 solution

Wikipedia 8-simplex coordinates . Actually, I computed the coordinates myself. The Wikipedia set is comparable.

If you can read Wolfram Mathematica code:

Table[ m=ConstantArray[0,{n+1,n}]; f=-(1/n); m[[1,1]]=1; m[[2;;n,1]]=f; Do[ m[[j,j]]=Sqrt[1-m[[j,All]].m[[j,All]]]; m[[j+1,j]]=q; m[[j+1,j]]=First[q/.Solve[m[[j]].m[[j+1]]==f]]; m[[j+1;;n,j]]=m[[j+1,j]];, {j,2,n} ]; m[[n+1]]=m[[n]]; m[[n+1,n]]=-m[[n+1,n]]; m, {n,8,8} ]

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