Definitions of Hill's Tetrahedra

type of Hill's tetrahedron:

1

2

3

height

labels

half turn

This Demonstration gives definitions of Hill's tetrahedra.

Details

Let ABC be an equilateral triangle and let the red lines be normals to the plane of the triangle. Let 1, 2, 3, and 4 be points on the normals at heights

0

,

h

,

2h

,

3h

. Then the tetrahedron with vertices 1, 2, 3, 4 is Hill's tetrahedron of type 1.

Let the points 5 and 6 be the midpoints of the segments 23 and 14. Then the straight line 56 is the axis of symmetry of the tetrahedron.

The tetrahedron with vertices 1, 2, 3, 6 is Hill's tetrahedron of type 2. Hill's tetrahedron of type 1 consists of two congruent Hill's tetrahedra of type 2.

The tetrahedron with vertices 1, 2, 4, 5 is Hill's tetrahedron of type 3. Hill's tetrahedron of type 1 consists of two congruent Hill's tetrahedra of type 3.

V. G. Boltyanskii, Tretja Problema Hilberta, Moscow: Nauka, 1977. Translated by R. A. Silverman as Hilbert's Third Problem (New York: John Wiley & Sons, Inc., 1978).

External Links

Hilbert's Problems (Wolfram MathWorld)

Tetrahedron (Wolfram MathWorld)

Permanent Citation

Izidor Hafner

"Definitions of Hill's Tetrahedra"
http://demonstrations.wolfram.com/DefinitionsOfHillsTetrahedra/
Wolfram Demonstrations Project
Published: March 7, 2011