How minimum positive integer dimension does it take?

Geometry Level 4

This problem’s question: {\color{#D61F06}\text{This problem's question:}} What is the minimum positive integer dimension it takes for the content (length for 1-dimensional, area for 2-dimensional, volume for 3-dimensional, etc.) of the regular n-simplex circumscribed by a n-dimensional unit radius ball to be less than 1% of the content of a unit sided R n \mathbb{R}^n cuboid (in 1 dimension,a line segment of length 2 (the centroid is in the middle); in 2 dimensions, a unit square, in 3 dimensions, a unit cube, etc.)?

A regular simplex has all edge lengths equal and all vertices at equal distance from the centroid of the figure. In the case of this problem the common distance from the centroid is 1 1 .


The answer is 6.

A Former Brilliant Member by A Former Brilliant Member

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1 solution

The answer is 6 by direct computation.

  1. ( 1 − 1 ) \left( \begin{array}{c} 1 \\ -1 \\ \end{array} \right) : content is 2.

  2. ( 1 0 − 1 2 3 2 − 1 2 − 3 2 ) \left( \begin{array}{cc} 1 & 0 \\ -\frac{1}{2} & \frac{\sqrt{3}}{2} \\ -\frac{1}{2} & -\frac{\sqrt{3}}{2} \\ \end{array} \right) : content is 1.29903810567666.

  3. ( 1 0 0 − 1 3 2 2 3 0 − 1 3 − 2 3 2 3 − 1 3 − 2 3 − 2 3 ) \left( \begin{array}{ccc} 1 & 0 & 0 \\ -\frac{1}{3} & \frac{2 \sqrt{2}}{3} & 0 \\ -\frac{1}{3} & -\frac{\sqrt{2}}{3} & \sqrt{\frac{2}{3}} \\ -\frac{1}{3} & -\frac{\sqrt{2}}{3} & -\sqrt{\frac{2}{3}} \\ \end{array} \right) : content is 0.513200239279667.

  4. ( 1 0 0 0 − 1 4 15 4 0 0 − 1 4 − 5 3 4 5 6 0 − 1 4 − 5 3 4 − 5 6 2 5 2 2 − 1 4 − 5 3 4 − 5 6 2 − 5 2 2 ) \left( \begin{array}{cccc} 1 & 0 & 0 & 0 \\ -\frac{1}{4} & \frac{\sqrt{15}}{4} & 0 & 0 \\ -\frac{1}{4} & -\frac{\sqrt{\frac{5}{3}}}{4} & \sqrt{\frac{5}{6}} & 0 \\ -\frac{1}{4} & -\frac{\sqrt{\frac{5}{3}}}{4} & -\frac{\sqrt{\frac{5}{6}}}{2} & \frac{\sqrt{\frac{5}{2}}}{2} \\ -\frac{1}{4} & -\frac{\sqrt{\frac{5}{3}}}{4} & -\frac{\sqrt{\frac{5}{6}}}{2} & -\frac{\sqrt{\frac{5}{2}}}{2} \\ \end{array} \right) : content is 0.145577342285143.

  5. ( 1 0 0 0 0 − 1 5 2 6 5 0 0 0 − 1 5 − 3 2 5 3 10 0 0 − 1 5 − 3 2 5 − 1 10 2 5 0 − 1 5 − 3 2 5 − 1 10 − 1 5 3 5 − 1 5 − 3 2 5 − 1 10 − 1 5 − 3 5 ) \left( \begin{array}{ccccc} 1 & 0 & 0 & 0 & 0 \\ -\frac{1}{5} & \frac{2 \sqrt{6}}{5} & 0 & 0 & 0 \\ -\frac{1}{5} & -\frac{\sqrt{\frac{3}{2}}}{5} & \frac{3}{\sqrt{10}} & 0 & 0 \\ -\frac{1}{5} & -\frac{\sqrt{\frac{3}{2}}}{5} & -\frac{1}{\sqrt{10}} & \frac{2}{\sqrt{5}} & 0 \\ -\frac{1}{5} & -\frac{\sqrt{\frac{3}{2}}}{5} & -\frac{1}{\sqrt{10}} & -\frac{1}{\sqrt{5}} & \sqrt{\frac{3}{5}} \\ -\frac{1}{5} & -\frac{\sqrt{\frac{3}{2}}}{5} & -\frac{1}{\sqrt{10}} & -\frac{1}{\sqrt{5}} & -\sqrt{\frac{3}{5}} \\ \end{array} \right) : content is 0.032199378875997.

  6. ( 1 0 0 0 0 0 − 1 6 35 6 0 0 0 0 − 1 6 − 7 5 6 14 15 0 0 0 − 1 6 − 7 5 6 − 7 30 2 7 2 2 0 0 − 1 6 − 7 5 6 − 7 30 2 − 7 2 6 7 3 0 − 1 6 − 7 5 6 − 7 30 2 − 7 2 6 − 7 6 7 3 2 − 1 6 − 7 5 6 − 7 30 2 − 7 2 6 − 7 6 − 7 3 2 ) \left( \begin{array}{cccccc} 1 & 0 & 0 & 0 & 0 & 0 \\ -\frac{1}{6} & \frac{\sqrt{35}}{6} & 0 & 0 & 0 & 0 \\ -\frac{1}{6} & -\frac{\sqrt{\frac{7}{5}}}{6} & \sqrt{\frac{14}{15}} & 0 & 0 & 0 \\ -\frac{1}{6} & -\frac{\sqrt{\frac{7}{5}}}{6} & -\frac{\sqrt{\frac{7}{30}}}{2} & \frac{\sqrt{\frac{7}{2}}}{2} & 0 & 0 \\ -\frac{1}{6} & -\frac{\sqrt{\frac{7}{5}}}{6} & -\frac{\sqrt{\frac{7}{30}}}{2} & -\frac{\sqrt{\frac{7}{2}}}{6} & \frac{\sqrt{7}}{3} & 0 \\ -\frac{1}{6} & -\frac{\sqrt{\frac{7}{5}}}{6} & -\frac{\sqrt{\frac{7}{30}}}{2} & -\frac{\sqrt{\frac{7}{2}}}{6} & -\frac{\sqrt{7}}{6} & \frac{\sqrt{\frac{7}{3}}}{2} \\ -\frac{1}{6} & -\frac{\sqrt{\frac{7}{5}}}{6} & -\frac{\sqrt{\frac{7}{30}}}{2} & -\frac{\sqrt{\frac{7}{2}}}{6} & -\frac{\sqrt{7}}{6} & -\frac{\sqrt{\frac{7}{3}}}{2} \\ \end{array} \right) : content is 0.00583521540441843.

  7. ( 1 0 0 0 0 0 0 − 1 7 4 3 7 0 0 0 0 0 − 1 7 − 2 7 3 2 5 21 0 0 0 0 − 1 7 − 2 7 3 − 2 105 4 2 35 0 0 0 − 1 7 − 2 7 3 − 2 105 − 2 35 6 7 0 0 − 1 7 − 2 7 3 − 2 105 − 2 35 − 2 21 4 21 0 − 1 7 − 2 7 3 − 2 105 − 2 35 − 2 21 − 2 21 2 7 − 1 7 − 2 7 3 − 2 105 − 2 35 − 2 21 − 2 21 − 2 7 ) \left( \begin{array}{ccccccc} 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ -\frac{1}{7} & \frac{4 \sqrt{3}}{7} & 0 & 0 & 0 & 0 & 0 \\ -\frac{1}{7} & -\frac{2}{7 \sqrt{3}} & 2 \sqrt{\frac{5}{21}} & 0 & 0 & 0 & 0 \\ -\frac{1}{7} & -\frac{2}{7 \sqrt{3}} & -\frac{2}{\sqrt{105}} & 4 \sqrt{\frac{2}{35}} & 0 & 0 & 0 \\ -\frac{1}{7} & -\frac{2}{7 \sqrt{3}} & -\frac{2}{\sqrt{105}} & -\sqrt{\frac{2}{35}} & \sqrt{\frac{6}{7}} & 0 & 0 \\ -\frac{1}{7} & -\frac{2}{7 \sqrt{3}} & -\frac{2}{\sqrt{105}} & -\sqrt{\frac{2}{35}} & -\sqrt{\frac{2}{21}} & \frac{4}{\sqrt{21}} & 0 \\ -\frac{1}{7} & -\frac{2}{7 \sqrt{3}} & -\frac{2}{\sqrt{105}} & -\sqrt{\frac{2}{35}} & -\sqrt{\frac{2}{21}} & -\frac{2}{\sqrt{21}} & \frac{2}{\sqrt{7}} \\ -\frac{1}{7} & -\frac{2}{7 \sqrt{3}} & -\frac{2}{\sqrt{105}} & -\sqrt{\frac{2}{35}} & -\sqrt{\frac{2}{21}} & -\frac{2}{\sqrt{21}} & -\frac{2}{\sqrt{7}} \\ \end{array} \right) : content is 0.00089554264510493.

  8. ( 1 0 0 0 0 0 0 0 − 1 8 3 7 8 0 0 0 0 0 0 − 1 8 − 3 8 7 3 3 7 2 0 0 0 0 0 − 1 8 − 3 8 7 − 3 7 4 15 4 0 0 0 0 − 1 8 − 3 8 7 − 3 7 4 − 3 5 4 3 10 0 0 0 − 1 8 − 3 8 7 − 3 7 4 − 3 5 4 − 3 4 10 3 3 2 4 0 0 − 1 8 − 3 8 7 − 3 7 4 − 3 5 4 − 3 4 10 − 3 2 4 3 2 0 − 1 8 − 3 8 7 − 3 7 4 − 3 5 4 − 3 4 10 − 3 2 4 − 3 4 3 4 − 1 8 − 3 8 7 − 3 7 4 − 3 5 4 − 3 4 10 − 3 2 4 − 3 4 − 3 4 ) \left( \begin{array}{cccccccc} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ -\frac{1}{8} & \frac{3 \sqrt{7}}{8} & 0 & 0 & 0 & 0 & 0 & 0 \\ -\frac{1}{8} & -\frac{3}{8 \sqrt{7}} & \frac{3 \sqrt{\frac{3}{7}}}{2} & 0 & 0 & 0 & 0 & 0 \\ -\frac{1}{8} & -\frac{3}{8 \sqrt{7}} & -\frac{\sqrt{\frac{3}{7}}}{4} & \frac{\sqrt{15}}{4} & 0 & 0 & 0 & 0 \\ -\frac{1}{8} & -\frac{3}{8 \sqrt{7}} & -\frac{\sqrt{\frac{3}{7}}}{4} & -\frac{\sqrt{\frac{3}{5}}}{4} & \frac{3}{\sqrt{10}} & 0 & 0 & 0 \\ -\frac{1}{8} & -\frac{3}{8 \sqrt{7}} & -\frac{\sqrt{\frac{3}{7}}}{4} & -\frac{\sqrt{\frac{3}{5}}}{4} & -\frac{3}{4 \sqrt{10}} & \frac{3 \sqrt{\frac{3}{2}}}{4} & 0 & 0 \\ -\frac{1}{8} & -\frac{3}{8 \sqrt{7}} & -\frac{\sqrt{\frac{3}{7}}}{4} & -\frac{\sqrt{\frac{3}{5}}}{4} & -\frac{3}{4 \sqrt{10}} & -\frac{\sqrt{\frac{3}{2}}}{4} & \frac{\sqrt{3}}{2} & 0 \\ -\frac{1}{8} & -\frac{3}{8 \sqrt{7}} & -\frac{\sqrt{\frac{3}{7}}}{4} & -\frac{\sqrt{\frac{3}{5}}}{4} & -\frac{3}{4 \sqrt{10}} & -\frac{\sqrt{\frac{3}{2}}}{4} & -\frac{\sqrt{3}}{4} & \frac{3}{4} \\ -\frac{1}{8} & -\frac{3}{8 \sqrt{7}} & -\frac{\sqrt{\frac{3}{7}}}{4} & -\frac{\sqrt{\frac{3}{5}}}{4} & -\frac{3}{4 \sqrt{10}} & -\frac{\sqrt{\frac{3}{2}}}{4} & -\frac{\sqrt{3}}{4} & -\frac{3}{4} \\ \end{array} \right) : content is 0.00011918204171317.

9. ( 1 0 0 0 0 0 0 0 0 − 1 9 4 5 9 0 0 0 0 0 0 0 − 1 9 − 5 18 35 6 0 0 0 0 0 0 − 1 9 − 5 18 − 5 7 6 2 5 21 0 0 0 0 0 − 1 9 − 5 18 − 5 7 6 − 5 21 3 5 3 3 0 0 0 0 − 1 9 − 5 18 − 5 7 6 − 5 21 3 − 1 3 3 2 2 3 0 0 0 − 1 9 − 5 18 − 5 7 6 − 5 21 3 − 1 3 3 − 1 3 2 5 6 0 0 − 1 9 − 5 18 − 5 7 6 − 5 21 3 − 1 3 3 − 1 3 2 − 5 6 3 2 5 3 3 0 − 1 9 − 5 18 − 5 7 6 − 5 21 3 − 1 3 3 − 1 3 2 − 5 6 3 − 5 3 3 5 3 − 1 9 − 5 18 − 5 7 6 − 5 21 3 − 1 3 3 − 1 3 2 − 5 6 3 − 5 3 3 − 5 3 ) \left( \begin{array}{ccccccccc} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ -\frac{1}{9} & \frac{4 \sqrt{5}}{9} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ -\frac{1}{9} & -\frac{\sqrt{5}}{18} & \frac{\sqrt{35}}{6} & 0 & 0 & 0 & 0 & 0 & 0 \\ -\frac{1}{9} & -\frac{\sqrt{5}}{18} & -\frac{\sqrt{\frac{5}{7}}}{6} & 2 \sqrt{\frac{5}{21}} & 0 & 0 & 0 & 0 & 0 \\ -\frac{1}{9} & -\frac{\sqrt{5}}{18} & -\frac{\sqrt{\frac{5}{7}}}{6} & -\frac{\sqrt{\frac{5}{21}}}{3} & \frac{5}{3 \sqrt{3}} & 0 & 0 & 0 & 0 \\ -\frac{1}{9} & -\frac{\sqrt{5}}{18} & -\frac{\sqrt{\frac{5}{7}}}{6} & -\frac{\sqrt{\frac{5}{21}}}{3} & -\frac{1}{3 \sqrt{3}} & \frac{2 \sqrt{2}}{3} & 0 & 0 & 0 \\ -\frac{1}{9} & -\frac{\sqrt{5}}{18} & -\frac{\sqrt{\frac{5}{7}}}{6} & -\frac{\sqrt{\frac{5}{21}}}{3} & -\frac{1}{3 \sqrt{3}} & -\frac{1}{3 \sqrt{2}} & \sqrt{\frac{5}{6}} & 0 & 0 \\ -\frac{1}{9} & -\frac{\sqrt{5}}{18} & -\frac{\sqrt{\frac{5}{7}}}{6} & -\frac{\sqrt{\frac{5}{21}}}{3} & -\frac{1}{3 \sqrt{3}} & -\frac{1}{3 \sqrt{2}} & -\frac{\sqrt{\frac{5}{6}}}{3} & \frac{2 \sqrt{\frac{5}{3}}}{3} & 0 \\ -\frac{1}{9} & -\frac{\sqrt{5}}{18} & -\frac{\sqrt{\frac{5}{7}}}{6} & -\frac{\sqrt{\frac{5}{21}}}{3} & -\frac{1}{3 \sqrt{3}} & -\frac{1}{3 \sqrt{2}} & -\frac{\sqrt{\frac{5}{6}}}{3} & -\frac{\sqrt{\frac{5}{3}}}{3} & \frac{\sqrt{5}}{3} \\ -\frac{1}{9} & -\frac{\sqrt{5}}{18} & -\frac{\sqrt{\frac{5}{7}}}{6} & -\frac{\sqrt{\frac{5}{21}}}{3} & -\frac{1}{3 \sqrt{3}} & -\frac{1}{3 \sqrt{2}} & -\frac{\sqrt{\frac{5}{6}}}{3} & -\frac{\sqrt{\frac{5}{3}}}{3} & -\frac{\sqrt{5}}{3} \\ \end{array} \right) : content is 0.0000140005686246944.

The formulas for the vertices of the regular simplices and the content of general, convex hull simplices are in the Wikipedia article.

As a point of interest and amusement, computing the vertices for all simplices of dimension 1 to 1000 took only a minute or so. When I tried to display those results, I consumed more than 24GB of memory and crashed Wolfram Mathematica by running it out of available memory.

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