A Theorem on the Dihedral Angles of a Tetrahedron

a

1.2

b

1

c

1

d

1.2

e

1.2

f

1.2

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a ⊗ e	b ⊗ f	c ⊗ d
2.94387	2.94387	2.94387

edge	length	angle	angle (radians)
a	1.2	α	1.69989
b	1.	β	1.92172
c	1.	γ	1.69989
d	1.2	δ	2.09151
e	1.2	ϵ	1.74
f	1.2	ϕ	2.09151

Let

a

,

b

,

c

,

d

,

e

,

f

be the edges of a tetrahedron and let

α

,

β

,

γ

,

δ

,

ϵ

,

ϕ

be the corresponding dihedral angles. Define the function

p⊗q

on pairs of opposite edges and dihedral angles as

2

p

+

2

q

+2pqcotp'cotq'

, where

p'

is the dihedral angle at

p

and

q'

is the dihedral angle at

q

.

A theorem states that this function is constant on the three pairs of opposite edges of a tetrahedron.