"The volume of a regular tetrahedron (triangular pyramid with unit edges) is exactly half the volume of a square pyramid with unit edges." True or false? --rwg
Son of a gun. On Thu, Sep 22, 2016 at 8:05 PM, Bill Gosper <billgosper@gmail.com> wrote:
"The volume of a regular tetrahedron (triangular pyramid with unit edges) is exactly half the volume of a square pyramid with unit edges." True or false? --rwg _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
"The volume of a regular tetrahedron (triangular pyramid with unit edges) is exactly half the volume of a square pyramid with unit edges."
This is (I think) equivalent to stating that the volume of an octahedron is four times the volume of a tetrahedron (unit edges assumed). I searched for an early observation of this and found an 1820 mention by Nathaniel John Larkin in his "An Introduction to Solid Geometry and to the study of Crystallography..." (page 103, free ebook at Google). Perhaps someone knows of an even earlier reference?
On Sun, Sep 25, 2016 at 12:01 PM, Hans Havermann <gladhobo@bell.net> wrote:
"The volume of a regular tetrahedron (triangular pyramid with unit edges) is exactly half the volume of a square pyramid with unit edges."
This is (I think) equivalent to stating that the volume of an octahedron is four times the volume of a tetrahedron (unit edges assumed).
Yes because you can slice an octahedron into two square pyramids. The geometrical "proof" of this is that you can assemble 4 unit tetrahedrons and a unit octahedron to produce a tetrahedron with edge length two, and therefore area equal to 8 tetrahedrons. I don't have the ability to draw a picture of this here, but start with an equilateral triangle of side 2, and divide it into 4 equilateral triangles of side 1. Place tetrahedrons on the 3 of these that are in the same orientation, and then add as a second layer a fourth tetrahedron whose bottom three vertices are the points of the first 3 tetrahedrons. These four tetrahedrons outline the shape of a side-2 tetrahedron, and the space in the large tetrahedron that isn't in any of the small ones is exactly a unit octahedron. Thanks to Kate Fractal for the idea of this construction. Andy
Kate Fractal should be invited to join math-fun! Jim Propp On Sunday, September 25, 2016, Andy Latto <andy.latto@pobox.com> wrote:
On Sun, Sep 25, 2016 at 12:01 PM, Hans Havermann <gladhobo@bell.net <javascript:;>> wrote:
"The volume of a regular tetrahedron (triangular pyramid with unit edges) is exactly half the volume of a square pyramid with unit edges."
This is (I think) equivalent to stating that the volume of an octahedron is four times the volume of a tetrahedron (unit edges assumed).
Yes because you can slice an octahedron into two square pyramids.
The geometrical "proof" of this is that you can assemble 4 unit tetrahedrons and a unit octahedron to produce a tetrahedron with edge length two, and therefore area equal to 8 tetrahedrons.
I don't have the ability to draw a picture of this here, but start with an equilateral triangle of side 2, and divide it into 4 equilateral triangles of side 1. Place tetrahedrons on the 3 of these that are in the same orientation, and then add as a second layer a fourth tetrahedron whose bottom three vertices are the points of the first 3 tetrahedrons. These four tetrahedrons outline the shape of a side-2 tetrahedron, and the space in the large tetrahedron that isn't in any of the small ones is exactly a unit octahedron.
Thanks to Kate Fractal for the idea of this construction.
Andy
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com <javascript:;> https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
participants (5)
-
Allan Wechsler -
Andy Latto -
Bill Gosper -
Hans Havermann -
James Propp