Simplex coordinate sets

For 1 dimensional to 9-dimensional simplices, here are coordinates sets for simplices circumscribed in a unit radius ball of the same dimension.

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

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

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

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

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

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

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

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