Editing Simplex (section)

=== Dihedral angles of the regular ''n''-simplex ===
Any two {{math|(''n'' − 1)}}-dimensional faces of a regular {{mvar|n}}-dimensional simplex are themselves regular {{math|(''n'' − 1)}}-dimensional simplices, and they have the same [[dihedral angle]] of {{math|cos<sup>−1</sup>(1/''n'')}}.<ref>{{cite journal | journal =American Mathematical Monthly | volume = 109 | issue = 8 | date = October 2002 | pages = 756–8 | title = An Elementary Calculation of the Dihedral Angle of the Regular {{mvar|n}}-Simplex | first1 = Harold R. | last1 = Parks | author-link = Harold R. Parks |first2 = Dean C. |last2=Wills | jstor = 3072403 | doi=10.2307/3072403}}</ref><ref>{{cite thesis |type=PhD | publisher = Oregon State University | date = June 2009 | title = Connections between combinatorics of permutations and algorithms and geometry |first1= Harold R. |last2=Parks |first2 = Dean C. |last1=Wills | url = http://ir.library.oregonstate.edu/xmlui/handle/1957/11929 |hdl=1957/11929}}</ref> 

This can be seen by noting that the center of the standard simplex is <math display="inline">\left(\frac{1}{n+1}, \dots, \frac{1}{n+1}\right)</math>, and the centers of its faces are coordinate permutations of <math display="inline">\left(0, \frac{1}{n}, \dots, \frac{1}{n}\right)</math>. Then, by symmetry, the vector pointing from <math display="inline">\left(\frac{1}{n+1}, \dots, \frac{1}{n+1}\right)</math> to <math display="inline">\left(0, \frac{1}{n}, \dots, \frac{1}{n}\right)</math> is perpendicular to the faces. So the vectors normal to the faces are permutations of <math>(-n, 1, \dots, 1)</math>, from which the dihedral angles are calculated.