Disphenoid

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Disphenoid tetrahedron.png
The tetragonal and digonal disphenoids can be positioned inside a cuboid bisecting two opposite faces. Both have four equal edges going around the sides. The digonal has two pairs of congruent isosceles triangle faces, while the tetragonal has four congruent isosceles triangle faces.
Rhombic disphenoid.png
A rhombic disphenoid has congruent scalene triangle faces, and can fit diagonally inside of a cuboid. It has three sets of edge lengths, existing as opposite pairs.

In geometry, a disphenoid (from Greek sphenoeides 'wedgelike') is a tetrahedron whose four faces are congruent acute-angled triangles. [1] It can also be described as a tetrahedron in which every two edges that are opposite each other have equal lengths. Other names for the same shape are isotetrahedron, [2] sphenoid, [3] bisphenoid, [3] isosceles tetrahedron, [4] equifacial tetrahedron, [5] almost regular tetrahedron, [6] and tetramonohedron. [7]

Contents

All the solid angles and vertex figures of a disphenoid are the same, and the sum of the face angles at each vertex is equal to two right angles. However, a disphenoid is not a regular polyhedron, because, in general, its faces are not regular polygons, and its edges have three different lengths.

Special cases and generalizations

If the faces of a disphenoid are equilateral triangles, it is a regular tetrahedron with Td tetrahedral symmetry, although this is not normally called a disphenoid. When the faces of a disphenoid are isosceles triangles, it is called a tetragonal disphenoid. In this case it has D2d dihedral symmetry. A sphenoid with scalene triangles as its faces is called a rhombic disphenoid and it has D2 dihedral symmetry. Unlike the tetragonal disphenoid, the rhombic disphenoid has no reflection symmetry, so it is chiral. [8] Both tetragonal disphenoids and rhombic disphenoids are isohedra: as well as being congruent to each other, all of their faces are symmetric to each other.

It is not possible to construct a disphenoid with right triangle or obtuse triangle faces. [4] When right triangles are glued together in the pattern of a disphenoid, they form a flat figure (a doubly-covered rectangle) that does not enclose any volume. [8] When obtuse triangles are glued in this way, the resulting surface can be folded to form a disphenoid (by Alexandrov's uniqueness theorem) but one with acute triangle faces and with edges that in general do not lie along the edges of the given obtuse triangles.

Two more types of tetrahedron generalize the disphenoid and have similar names. The digonal disphenoid has faces with two different shapes, both isosceles triangles, with two faces of each shape. The phyllic disphenoid similarly has faces with two shapes of scalene triangles.

Disphenoids can also be seen as digonal antiprisms or as alternated quadrilateral prisms.

Characterizations

A tetrahedron is a disphenoid if and only if its circumscribed parallelepiped is right-angled. [9]

We also have that a tetrahedron is a disphenoid if and only if the center in the circumscribed sphere and the inscribed sphere coincide. [10]

Another characterization states that if d1, d2 and d3 are the common perpendiculars of AB and CD; AC and BD; and AD and BC respectively in a tetrahedron ABCD, then the tetrahedron is a disphenoid if and only if d1, d2 and d3 are pairwise perpendicular. [9]

The disphenoids are the only polyhedra having infinitely many non-self-intersecting closed geodesics. On a disphenoid, all closed geodesics are non-self-intersecting. [11]

The disphenoids are the tetrahedra in which all four faces have the same perimeter, the tetrahedra in which all four faces have the same area, [10] and the tetrahedra in which the angular defects of all four vertices equal π. They are the polyhedra having a net in the shape of an acute triangle, divided into four similar triangles by segments connecting the edge midpoints. [6]

Metric formulas

The volume of a disphenoid with opposite edges of length l, m and n is given by: [12]

The circumscribed sphere has radius [12] (the circumradius):

and the inscribed sphere has radius: [12]

where V is the volume of the disphenoid and T is the area of any face, which is given by Heron's formula. There is also the following interesting relation connecting the volume and the circumradius: [12]

The squares of the lengths of the bimedians are: [12]

Other properties

If the four faces of a tetrahedron have the same perimeter, then the tetrahedron is a disphenoid. [10]

If the four faces of a tetrahedron have the same area, then it is a disphenoid. [9] [10]

The centers in the circumscribed and inscribed spheres coincide with the centroid of the disphenoid. [12]

The bimedians are perpendicular to the edges they connect and to each other. [12]

Honeycombs and crystals

A space-filling tetrahedral disphenoid inside a cube. Two edges have dihedral angles of 90deg, and four edges have dihedral angles of 60deg. Oblate tetrahedrille cell.png
A space-filling tetrahedral disphenoid inside a cube. Two edges have dihedral angles of 90°, and four edges have dihedral angles of 60°.

Some tetragonal disphenoids will form honeycombs. The disphenoid whose four vertices are (-1, 0, 0), (1, 0, 0), (0, 1, 1), and (0, 1, -1) is such a disphenoid. [13] [14] Each of its four faces is an isosceles triangle with edges of lengths √3, √3, and 2. It can tessellate space to form the disphenoid tetrahedral honeycomb. As Gibb (1990) describes, it can be folded without cutting or overlaps from a single sheet of a4 paper. [15]

"Disphenoid" is also used to describe two forms of crystal:

Other uses

Six tetragonal disphenoids attached end-to-end in a ring construct a kaleidocycle, a paper toy that can rotate on 4 sets of faces in a hexagon. The rotation of the six disphenoids with opposite edges of length l, m and n (without loss of generality n≤l, n≤m) is physically realizable if and only if [16]

See also

Related Research Articles

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<span class="mw-page-title-main">Cuboctahedron</span> Polyhedron with 8 triangular faces and 6 square faces

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<span class="mw-page-title-main">Kite (geometry)</span> Quadrilateral symmetric across a diagonal

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<span class="mw-page-title-main">Truncated tetrahedron</span> Archimedean solid with 8 faces

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<span class="mw-page-title-main">Trapezohedron</span> Polyhedron made of congruent kites arranged radially

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<span class="mw-page-title-main">Snub disphenoid</span> Convex polyhedron with 12 triangular faces

In geometry, the snub disphenoid is a convex polyhedron with 12 equilateral triangles as its faces. It is an example of deltahedron and Johnson solid. It can be constructed in different approaches. This shape is also called Siamese dodecahedron, triangular dodecahedron, trigonal dodecahedron, or dodecadeltahedron.

<span class="mw-page-title-main">Cubic honeycomb</span> Only regular space-filling tessellation of the cube

The cubic honeycomb or cubic cellulation is the only proper regular space-filling tessellation in Euclidean 3-space made up of cubic cells. It has 4 cubes around every edge, and 8 cubes around each vertex. Its vertex figure is a regular octahedron. It is a self-dual tessellation with Schläfli symbol {4,3,4}. John Horton Conway called this honeycomb a cubille.

<span class="mw-page-title-main">Bitruncated cubic honeycomb</span>

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<span class="mw-page-title-main">Tetragonal disphenoid honeycomb</span>

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<span class="mw-page-title-main">Truncated 5-cell</span>

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<span class="mw-page-title-main">Regular dodecahedron</span> Convex polyhedron with 12 regular pentagonal faces

A regular dodecahedron or pentagonal dodecahedron is a dodecahedron composed of regular pentagonal faces, three meeting at each vertex. It is an example of Platonic solids, described as cosmic stellation by Plato in his dialogues, and it was used as part of Solar System proposed by Johannes Kepler. However, the regular dodecahedron, including the other Platonic solids, has already been described by other philosophers since antiquity.

<span class="mw-page-title-main">Compound of two tetrahedra</span> Polyhedral compound

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<span class="mw-page-title-main">Ideal polyhedron</span> Shape in hyperbolic geometry

In three-dimensional hyperbolic geometry, an ideal polyhedron is a convex polyhedron all of whose vertices are ideal points, points "at infinity" rather than interior to three-dimensional hyperbolic space. It can be defined as the convex hull of a finite set of ideal points. An ideal polyhedron has ideal polygons as its faces, meeting along lines of the hyperbolic space.

References

  1. ↑ Coxeter, H. S. M. (1973), Regular Polytopes (3rd ed.), Dover Publications, p.  15, ISBN   0-486-61480-8
  2. ↑ Akiyama, Jin; Matsunaga, Kiyoko (2020), "An Algorithm for Folding a Conway Tile into an Isotetrahedron or a Rectangle Dihedron", Journal of Information Processing , 28 (28): 750–758, doi: 10.2197/ipsjjip.28.750 , S2CID   230108666 .
  3. 1 2 Whittaker, E. J. W. (2013), Crystallography: An Introduction for Earth Science (and other Solid State) Students, Elsevier, p. 89, ISBN   9781483285566 .
  4. 1 2 Leech, John (1950), "Some properties of the isosceles tetrahedron", The Mathematical Gazette , 34 (310): 269–271, doi:10.2307/3611029, JSTOR   3611029, MR   0038667, S2CID   125145099 .
  5. ↑ Hajja, Mowaffaq; Walker, Peter (2001), "Equifacial tetrahedra", International Journal of Mathematical Education in Science and Technology, 32 (4): 501–508, doi:10.1080/00207390110038231, MR   1847966, S2CID   218495301 .
  6. 1 2 Akiyama, Jin (2007), "Tile-makers and semi-tile-makers", American Mathematical Monthly , 114 (7): 602–609, doi:10.1080/00029890.2007.11920450, JSTOR   27642275, MR   2341323, S2CID   32897155 .
  7. ↑ Demaine, Erik; O'Rourke, Joseph (2007), Geometric Folding Algorithms, Cambridge University Press, p. 424, ISBN   978-0-521-71522-5 .
  8. 1 2 Petitjean, Michel (2015), "The most chiral disphenoid" (PDF), MATCH Communications in Mathematical and in Computer Chemistry, 73 (2): 375–384, MR   3242747 .
  9. 1 2 3 Andreescu, Titu; Gelca, Razvan (2009), Mathematical Olympiad Challenges (2nd ed.), Birkhäuser, pp. 30–31.
  10. 1 2 3 4 Brown, B. H. (April 1926), "Theorem of Bang. Isosceles tetrahedra", Undergraduate Mathematics Clubs: Club Topics, American Mathematical Monthly , 33 (4): 224–226, doi:10.1080/00029890.1926.11986564, JSTOR   2299548 .
  11. ↑ Fuchs, Dmitry [in German]; Fuchs, Ekaterina (2007), "Closed geodesics on regular polyhedra" (PDF), Moscow Mathematical Journal, 7 (2): 265–279, 350, doi:10.17323/1609-4514-2007-7-2-265-279, MR   2337883 .
  12. 1 2 3 4 5 6 7 Leech, John (1950), "Some properties of the isosceles tetrahedron", Mathematical Gazette, 34 (310): 269–271, doi:10.2307/3611029, JSTOR   3611029, S2CID   125145099 .
  13. ↑ Coxeter (1973 , pp. 71–72).
  14. ↑ Senechal, Marjorie (1981), "Which tetrahedra fill space?", Mathematics Magazine , 54 (5): 227–243, doi:10.2307/2689983, JSTOR   2689983, MR   0644075
  15. ↑ Gibb, William (1990), "Paper patterns: solid shapes from metric paper", Mathematics in School, 19 (3): 2–4 Reprinted in Pritchard, Chris, ed. (2003), The Changing Shape of Geometry: Celebrating a Century of Geometry and Geometry Teaching, Cambridge University Press, pp. 363–366, ISBN   0-521-53162-4
  16. ↑ Sloane, N. J. A. (ed.), "SequenceA338336", The On-Line Encyclopedia of Integer Sequences , OEIS Foundation{{cite web}}: CS1 maint: overridden setting (link)
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