¶ Octahedron: Volume of Cube as Three (3)
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Index Entry
Octahedron: Volume of Cube as Three:
“I am going to take these four one-eighth octahedra, and because each has an equilateral triangular face, I can superimpose it on the equilateral triangular faces of a regular tetrahedron. There are four faces, so the four faces of the tetrahedra will accommodate four one-eighth octahedra superimposed so that the equilateral triangle faces are becoming congruent. This leaves a 90-degree angle sticking outwardly, and it makes the cube. So the cube then is one regular tetrahedron, with four one-eighth octahedra superimposed on its surface. The volume of a one-eighth octahedron is one-half, and four times one-half on the faces (4 x ½ = 2) and so the volume of the tetrahedron which I superimposed is one, so (2 + 1 = 3), so the volume of a cube in the system is three. That gets to be very interesting. Here is another nice whole number. You have the tetrahedron as one, octahedron is four, and cube is three. A cube as three is not what people have been thinking. They have been thinking that a cube was one, but I am using unity where a tetrahedron is one.”
¶ Citations
- Oregon Lecture #6, pp. 215-216. 10 Jul’62
