Definitions of Hill's Tetrahedra

This Demonstration gives definitions of Hill's tetrahedra.



  • [Snapshot]
  • [Snapshot]
  • [Snapshot]
  • [Snapshot]


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 , , , . 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).
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.

Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-Step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2018 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+