Deltoidal icositetrahedron

3D view of a deltoidal icositetrahedron ( animation )

Construction of the deltoid on the rhombic cuboctahedron

This triple-cut cube is topologically equivalent to the deltoidal icositetrahedron

The deltoidal icositetrahedron (also deltoidal icositetrahedron is called), a convex Ikositetraeder, so a polyhedron with side faces 24, in which said surfaces mutually congruent deltoids are. It is one of the Catalan bodies . It is dual to the diamond cuboctahedron and has 26 corners and 48 edges.

In crystallography and mineralogy , the deltoidal icositetrahedron is often (shortened) referred to only as an icositetrahedron , and also as a trapezoid or leucitohedron (it is the typical crystal form of leucite ).

Emergence

If square and triangular pyramids with the flank length and are placed on the 14 boundary surfaces of a cuboctahedron , a general deltoidal icosity tetrahedron is created if and are. The inscribed cuboctahedron has the edge length (i.e. a diagonal of the dragon's square , see below). ${\ displaystyle a}$ ${\ displaystyle b}$ ${\ displaystyle a> {\ tfrac {e} {2}} {\ sqrt {2}}}$ ${\ displaystyle b> {\ tfrac {e} {3}} {\ sqrt {3}}}$ ${\ displaystyle e}$

By connecting the centers of four edges that meet in every corner of the room with the rhombic cuboctahedron , a trapezium is created , the circumference of which is also the inscribed circle of the deltoid, the boundary surface of the deltoidal icosity tetrahedron. With this special type, all face angles (≈ 138 ° 7 '5 ") are of the same size and there is a uniform edge spherical radius .

Let be the edge length of the diamond cuboctahedron, then the resulting side lengths of the deltoid are given by

{\ displaystyle d}

{\ displaystyle a = d \, {\ sqrt {4-2 {\ sqrt {2}}}}}

{\ displaystyle b = \, {\ frac {2} {7}} \, d \, {\ sqrt {10 - {\ sqrt {2}}}}}

The side lengths of the deltoid are therefore in the following relationship:

{\ displaystyle (4 + {\ sqrt {2}}) \, a = 7b}

This special (regular) deltoidal icosity tetrahedron is the circumscribed body of three regular octagons (with edge length ) standing perpendicular to each other , which intersect at their corners.

{\ displaystyle a}

Furthermore, the deltoidalikositetrahedron can be seen as a triple-cut "inflated" cube , which is topologically equivalent with its 24 square boundary surfaces .

Related polyhedra

Dual body: rhombic cuboctahedron
Inscribed cube
Inscribed octahedron
Inscribed cuboctahedron

Formulas for the regular deltoidal icositetrahedron

For the deltoid

Sizes in the deltoid - What is remarkable about this dragon square, which is also a tangent square, is the fact that 3 of the 4 interior angles are the same size.

Sizes of the dragon square
Aspect ratio	${\ displaystyle b = {\ frac {1} {7}} (4 + {\ sqrt {2}}) \, a}$
Area	${\ displaystyle A = {\ frac {a ^ {2}} {14}} {\ sqrt {61 + 38 {\ sqrt {2}}}}}$
Inscribed radius	${\ displaystyle r = {\ frac {a} {2}} {\ sqrt {\ frac {7 + 4 {\ sqrt {2}}} {17}}}}$
1st diagonal	${\ displaystyle e = {\ frac {a} {2}} {\ sqrt {4 + 2 {\ sqrt {2}}}}}$
2nd diagonal	${\ displaystyle f = {\ frac {a} {7}} {\ sqrt {46 + 15 {\ sqrt {2}}}}}$
Pointed angle (3) ≈ 81 ° 34 ′ 44 ″	${\ displaystyle \ cos \, \ alpha = {\ frac {1} {4}} \, (2 - {\ sqrt {2}})}$
Obtuse angle (1) ≈ 115 ° 15 ′ 47 ″	${\ displaystyle \ cos \, \ beta = - {\ frac {1} {8}} \, (2 + {\ sqrt {2}})}$

For the polyhedron

Network of the deltoidal icosity tetrahedron

Sizes of a regular Deltoidikositetrahedron with edge length a or b
Volume ≈ 6.9a ³ ≈ 14.91b ³	${\ displaystyle V = {\ frac {2} {7}} \, a ^ {3} {\ sqrt {292 + 206 {\ sqrt {2}}}} = b ^ {3} {\ sqrt {122+ 71 {\ sqrt {2}}}}}$
Surface area ≈ 18.36a ² ≈ 30.69b ²	${\ displaystyle A_ {O} = {\ frac {12} {7}} \, a ^ {2} {\ sqrt {61 + 38 {\ sqrt {2}}}} = 6 \, b ^ {2} {\ sqrt {29-2 {\ sqrt {2}}}}}$
Inc sphere radius	${\ displaystyle \ rho = a \, {\ sqrt {\ frac {22 + 15 {\ sqrt {2}}} {34}}} = {\ frac {b} {2}} {\ sqrt {\ frac { 78 + 47 {\ sqrt {2}}} {17}}}}$
Edge ball radius	${\ displaystyle r = {\ frac {a} {2}} \ left (1 + {\ sqrt {2}} \ right) = {\ frac {b} {4}} \ left (2 + 3 {\ sqrt {2}} \ right)}$
Face angle ≈ 138 ° 7 ′ 5 ″	${\ displaystyle \ cos \, \ alpha = - {\ frac {1} {17}} \, (7 + 4 {\ sqrt {2}})}$
3D edge angle = 135 °	${\ displaystyle \ cos \, \ gamma = - {\ frac {1} {2}} {\ sqrt {2}}}$

Occurrence

In nature z. B. leucite , analcime and spessartine, preferably in the form of deltoidalikositetrahedra. Deltoidalikositetrahedra are also found in crystal form in other minerals of the garnet group or in fluorite . The deltoidalikositetrahedron, that is the form {hll} (with h> l), is either a special form of the crystal class m 3 m, a limiting shape of the pentagonicositetrahedron in the crystal class 432 or a limiting shape of the disdodecahedron in the crystal class m 3 .

Remarks

↑ With a the longer of the two sides was called.

Web links

Commons : Deltoidalikositetrahedron - collection of images, videos and audio files

Wiktionary: Deltoidalikositetrahedron - explanations of meanings, word origins, synonyms, translations

Eric W. Weisstein : Deltoidalikositetrahedron . In: MathWorld (English).

[1] With a the longer of the two sides was called.