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The first few tetrahedral numbers are

Tetrahedral numbers can be modelled in physical space. For example, the tetrahedral number 35 can be modelled with 35 billiard balls and the standard triangular billiards ball frame that holds 15 balls in place. Then 10 more balls are stacked on top of those, then another 6, then another three and one ball at the top completes the tetrahedron.

The parity of tetrahedral numbers follows the pattern odd-even-even-even.

In 1878, A.J. Meyl proved that there are only three tetrahedral numbers that are also square, namely, 1, 4 and 19600. So far, the only known tetrahedral number that is also a square pyramidal number is 1.

Tetrahedral numbers are found in the fourth position either from left to right or right to left in Pascal's triangle1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 The first six rows of Pascal's triangle In mathematics, Pascal's triangle is a geometric arrangement of the binomial coefficients in a triangle. In simple terms, Pascal's triangle can be constructed in the follo.

The tetrahedron with basic length 4 (summing up to 20) can be looked at as the 3D-representative of the tektraktys - which is the 4th triangular number (summing up to 10). The tektraktys was considered as holy by the PythagoreansThe Pythagoreans were an Hellenic organization of astronomers, musicians, mathematicians, and philosophers; who believed that all things are, essentially, numeric. The group strove to keep the discovery of irrational numbers a secret; and legends tell of. It can be shown that with tetrahedrons up to basic length 4 a Closest Sphere PackingIn mathematics, sphere packing problems are problems concerning arrangements of non-overlapping identical spheres which fill a space. Usually the space involved is three- dimensional Euclidean space. However, sphere packing problems can be generalised to filling leaves no gaps: http://www.pisquaredoversix.force9.co.uk/Tetrahedra.htm