Topic
Disphenoid
About: Disphenoid is a research topic. Over the lifetime, 39 publications have been published within this topic receiving 191 citations. The topic is also known as: isosceles tetrahedron & equifacial tetrahedron.
Papers
TL;DR: In this article, a tetrahedron having two right angles at each of two vertices is dissected into three smaller orthoschemes, and two cutting planes meet three of the four faces along lines which can easily be described.
Abstract: A tetrahedron having two right angles at each of two vertices was investigated by Lobachevsky (who called it a “pyramid”), Schlafli (who called it an “orthoscheme”), Wythoff (who called it “double-rectangular”), and Schoute (who called its theory “polygonometry”). There is a simple procedure for dissecting such a tetrahedron into three smaller orthoschemes. The two cutting planes meet three of the four faces (which are right-angled triangles) along lines which can easily be described. When the tetrahedron is unfolded so as to put all the faces in one plane, the arrangement of lines suggests an interesting theorem of absolute geometry. When a particular spherical orthoscheme of known volume is dissected into three pieces, and the volumes of these smaller orthoschemes are expressed as definite integrals, the result is a peculiar identity which has not been verified directly. There is a one-parameter family of orthoschemes for which the three smaller orthoschemes are all congruent; the Euclidean member of this family turns out to be related to a very simply frieze pattern of integers.
20 citations
TL;DR: Using the SternBrocot tree to explore the recursive structure of geodesics between vertices on a cube, it is proved that there are twice as many geodesic between certain pairs of vertices than other pairs.
16 citations
TL;DR: In this paper, the authors solved the problem of finding the expected volume of a daughter tetrahedron whose vertices are chosen at random (i.e. independently and uniformly) in the interior of a parent tetrahedral of unit volume.
Abstract: We solve a problem proposed by V. Klee (1969). He asked for a calculation of K, the expected value of V, the volume of a daughter tetrahedron whose vertices are chosen at random (i.e. independently and uniformly) in the interior of a parent tetrahedron of unit volume. We discover:
15 citations
Patent•
5 Aug 1999
TL;DR: In this paper, the polyhedron blocks of each set are fitted together in a corresponding transparent case to form the octahedron or tetrahedron, according to one of three levels of difficulty.
Abstract: Multi-polyhedral puzzles characterized by four tetrahedra and one octahedron which are divided into different sets of multiple polyhedron blocks having various configurations, each block having multiple faces and each face being one of several colors. In a preferred embodiment the polyhedron blocks of each octahedron or tetrahedron set are fitted together in a corresponding transparent case to form the octahedron or tetrahedron, according to one of three levels of difficulty. At the most advanced level of difficulty in assembling each octahedron and tetrahedron, the polyhedron blocks of each set are fitted together such that abutting faces of adjacent polyhedron blocks match in color and a prescribed color pattern is formed on the respective faces of the assembled octahedron or tetrahedron. At an intermediate level of difficulty, the polyhedron blocks are fitted together to form the prescribed color pattern on the faces of the assembled octahedron or tetrahedron without regard to matching colors of abutting polyhedron block faces. At an elementary level of difficulty, the polyhedron blocks are fitted together to form the corresponding octahedron or tetrahedron without regard to matching colors of abutting polyhedron block faces or formation of the prescribed color pattern on the faces of the assembled octahedron or tetrahedron. The assembled tetrahedra can be arranged on respective faces of the assembled octahedron to form a large tetrahedron, for packaging or storage purposes.
12 citations
TL;DR: It is shown that there are convex developments of a regular tetrahedron having the following shapes: an equilateral triangle, an isosceles triangle, a right-angled triangle, scalene triangles, rectangles, parallelograms, trapezoids, quadrilaterals which are not trapezoid, pentagons and hexagons.
Abstract: The best-known developments of a regular tetrahedron are an equilateral triangle and a parallelogram. Are there any other convex developments of a regular tetrahedron? In this paper we will show that there are convex developments of a regular tetrahedron having the following shapes: an equilateral triangle, an isosceles triangle, a right-angled triangle, scalene triangles, rectangles, parallelograms, trapezoids, quadrilaterals which are not trapezoids, pentagons and hexagons, and furthermore these cases exhaust all the possibilities of convex developments with sides n= =7. Here, we mean by a development of a polyhedron a connected plane figure, from which one can construct the polyhedron by folding it without getting overlap or gap. In so folding we do not require that the sides of the development should end up as the edges of the polyhedron.
11 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2017 | 1 |
| 2016 | 1 |
| 2015 | 2 |
| 2014 | 3 |
| 2013 | 1 |
| 2012 | 1 |