Editing Pentatope number (section)

{{short description|Number in the 5th cell of any row of Pascal's triangle}}
{{Pascal_triangle_simplex_numbers.svg|2=pentatope numbers}}

In [[number theory]], a '''pentatope number''' is a number in the fifth cell of any row of [[Pascal's triangle]] starting with the 5-term row {{nowrap|1 4 6 4 1}}, either from left to right or from right to left. It is named because it represents the number of 3-dimensional [[unit sphere]]s which can be [[Sphere packing|packed]] into a [[pentatope]] (a 4-dimensional [[tetrahedron]]) of increasing side lengths.

The first few numbers of this kind are:

: [[1 (number)|1]], [[5 (number)|5]], [[15 (number)|15]], [[35 (number)|35]], [[70 (number)|70]], [[126 (number)|126]], [[210 (number)|210]], [[330 (number)|330]], [[495 (number)|495]], [[715 (number)|715]], [[1001 (number)|1001]], [[1365  (number)|1365]] {{OEIS|id=A000332}}

[[Image:Pentatope of 70 spheres animation.gif|frame|right|A [[pentatope]] with side length 5 contains 70 [[3-sphere]]s.  Each layer represents one of the first five [[tetrahedral number]]s.  For example, the bottom (green) layer has 35 [[sphere]]s in total.]]

Pentatope numbers belong to the class of [[figurate number]]s, which can be represented as regular, discrete geometric patterns.<ref>{{citation|title=Figurate Numbers|first1=Elena|last1=Deza|author1-link=Elena Deza|first2=M.|last2=Deza|author2-link=Michel Deza|publisher=World Scientific|year=2012|isbn=9789814355483|page=162|contribution=3.1 Pentatope numbers and their multidimensional analogues}}</ref>