Aristotle Gap

Author

Eric W. Weisstein
September 26, 2023

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Figure

Graphics3D[{Text[Style["θ",Directive[24,FontFamily->"Times",Black]],{0,0,.9}],Opacity[.4],PolyhedronData[{"TetrahedralRing",5},"Polyhedron"]},Boxed->False,ViewPoint->{0,-2,1}]

5-ring ("Aristotle") gap

Mysteries in Packing Regular Tetrahedra
Jeffrey C. Lagarias and Chuanming Zong
1540–1549
Notices of the AMS Volume 59, Number 11
DOI: http://dx.doi.org/10.1090/noti918

Aristotle’s Error
In the history of mathematics, one of the earliest recorded mistakes was made by Aristotle. More than 2,300 years ago, Aristotle (384–322 BCE) taught that the regular tetrahedra fill space. In De Caelo (On the Heavens) Book III.8 he says (in translation): “Among surfaces it is agreed that there are three figures which fill the place that contain them—the triangle, the square and the hexagon: among solids only two, the pyramid and the cube” [3, 306b, p. 319]. Here “pyramid” refers to the regular tetrahedron, a Platonic solid. Thus Aristotle’s assertion can be taken to mean: space can be tiled by congruent regular tetrahedra.

Such a tiling is impossible. Let T denote a regular tetrahedron with unit edges and let α denote one of its dihedral angles. By a routine computation one can deduce that
α = arccos(1/3) ≈ 70◦32′.
If five tetrahedra are fitted around an edge (see Figure 1), then there is a small gap whose angular measure θ satisfies
0<θ=2π−5α≈7◦21′ <α,
and we conclude that regular tetrahedra cannot fill the space when arranged face-to-face

c=Mean[vnew[[{3,4}]]]

VectorAngle[vnew[[7]]-c,vnew[[8]]-c]//FullSimplify

N[%]

2π-5ArcCos[(1/3)]//N

N[ArcCos[(241/243)],10]

N[ArcCos[(241/243)]/Degree,10]

FullSimplify[Subtract@@{2π-5ArcCos[(1/3)],ArcCos[(241/243)]}]

N[Subtract@@{2π-5ArcCos[(1/3)],ArcCos[(241/243)]},10^5]