Platonic solid

The geometry is called the Platonic solids perfectly regular polyhedra ( polyhedra are three-dimensional body, the polygons ( polygons ) are limited as faces). The Platonic solids are named after the Greek philosopher Plato.

Clearly it is not possible for Platonic Solids to distinguish any two vertices, edges and faces only because of relations to other points of the polyhedron from each other. When you remove the indistinguishability of the corners, it is called regular polyhedra and thus includes the Kepler - Poinsot body. Waived If, however, on the indistinguishability of the surfaces and edges, it is called Archimedean bodies.

There are five types of platonic bodies: tetrahedron, hexahedron (cube, cube ), octahedron, dodecahedron, and icosahedron (that is, each Platonic solids is to exactly one regular copy of these five similar). Enter your name in Greek, the number of its faces again (4, 6, 8, 12 or 20).

• 3.1 Platonic solids as regular tilings of the sphere
• 3.2 derived from the Platonic solids polyhedra 3.2.1 Einbeschreibungen
• 3.2.2 Deadened platonic solids
• 3.2.3 star body

Alternative definitions

• Consider the symmetry figures, which reflect a polyhedron by means of rotations, reflections and translations on itself. The Platonic solids are then precisely those polyhedra, so there is for any pair of points or of faces or edges and such symmetry mapping that maps to. The Platonic solids are thus polyhedron of maximum symmetry.
• The Platonic solids are precisely those convex polyhedron whose faces are all congruent to each other ( congruent) regular polygons and an equal number of which collide in every corner. Without the Konvexheit you turn passes to the regular polyhedra.

Basic Properties

Platonic solids have the following properties:

• The surface is composed of surfaces, so they are polyhedra.
• You are convex: There are no reentrant corners or edges.
• The edges are all the same length.
• The faces are all congruent to each other, that is, they can be converted by rotations and translations into each other.
• All corners have the same surface and edge angle, all faces are equilateral and equiangular.
• All vertices have the same distance from the center.
• Due to the symmetry of vertices, edges and faces, there is a circumscribed sphere, an edge and a ball inscribed ball.
• They are either tetrahedron, hexahedron, octahedron, dodecahedron or icosahedron.

Types

To Umkugelvolumen

Number

Two Platonic solids of the same type are similar to each other, that is, a Platonic solid is already determined by the specification of a single size, for example, edge length, body volume or Umkugelradius, clearly. In this sense it is therefore justified to speak of the tetrahedron, hexahedron, etc. the.

Under the conditions that the surface consists only of equal and regular polygons that meet the same number of edges at each corner and the body is convex (free from indentations ), there are exactly five Platonic solids. The proof of this can already be found in Euclid. It is based on the following considerations:

• For each polyhedron, the sum of the inner angles of all adjacent areas is smaller than 360 °. Had it exactly 360 °, the surfaces would lie in a plane; even at more than 360 ° no corner would be possible.
• On the other hand, at each corner of a polyhedron must meet at least three areas.

So, are one body in all faces equilateral triangles ( interior angle 60 °), so therefore can meet at a corner three, four or five triangles (sum of angles 180 °, 240 °, 300 °). Are the side surfaces squares ( interior angle 90 °) or regular pentagons ( interior angle 108 ° ), so you can meet them each three (sum of angles 270 ° and 324 ° in squares with pentagons ). Six equilateral triangles, four squares and three regular hexagons ( interior angle 120 °) each give exactly 360 °, so that no corner of the room is created, but a regular tiling of the plane. All other possibilities ( four regular pentagons, three regular heptagons, etc.) exceed this angle already:

In three or four equilateral triangles and three squares at each corner is easy to see that there is indeed appropriate body. The icosahedron and dodecahedron is not immediately clear that the polygons come together without gaps and overlaps. To prove this, serve the following considerations: An icosahedron - in which five equilateral triangles collide in a corner - you can construct as follows:

It connects with two pentagons that are parallel to each other and which are rotated against each other, each of the " twisted " corners together so that ten equilateral triangles (expressed formally: You form a pentagon, an anti- prism). If, on the base and on the top surface of each of a five-sided pyramid ( with five equilateral triangles as a jacket ), so you get a body with 12 corners and 20 equilateral triangles. It can be seen (eg by recalculation) that the two corresponding to the pyramid tips corners and the ten corners of the antiprism congruent ( with the same dihedral angles ) are so actually a completely regular ( a regular ) polyhedra present. The dodecahedron is then the dual polyhedron. (Without this consideration, it is not self-evident that the dodecahedron can be actually realized by planar pentagons. )

The five Platonic solids are shown above ie (up to similarity ) is in fact the only convex bodies of this type ( congruent regular faces, congruent corners - the regularity need not be provided ).

( A complete proof under somewhat weaker assumptions - for spherical polyhedra - can be done with the Euler's polyhedron formula. )

In short: On a corner three, four or five equilateral triangles can come together. Also, three squares, or three regular pentagons are possible. Other options are not available.

More polyhedron with regular polygons as faces arise only if polygons with different number of vertices (but possibly the same Eckenart ) be allowed - to include the Archimedean body - and the body in which do not coincide at each corner the same number of polygons.

Duality

Mathematical duality generally prevails between a convex polyhedron and its so-called dual body. The edges thereof are constructed by connecting the centers of the respectively adjacent side faces of the polyhedron each other.

Thus, the dual polyhedron has just as many corners as the Ausgangspolyeder surfaces has. Dual body also has the same number of surfaces as the output member has corners. The latter can be thought of as spatially, that each ( " enlarged" ) surface of the dual body cuts off a corner of the main body. Third is that the Dualpolyeder and his Ausgangspolyeder have the same number of edges. This can also be seen from the above construction: two " adjacent side surfaces " together form an edge of the Ausgangspolyeders, and the " connection of the two center points " of this adjacent side surfaces represents an edge of the dual body

In the Platonic solids, as a subgroup of convex polyhedra, there are dual with respect to their body following features: First, here are the output and dual body the same geometric center. Secondly, the dual body of a Platonic body also itself a Platonic solid. Here, hexahedron (cube ), and octahedron and dodecahedron, and icosahedron respectively form a dual pair. The tetrahedron is dual to itself, but with the dual tetrahedron is in ( scaled ) centrally symmetrical position, that is, it " upside down ". Third: If you repeat the above construction and designed for the dual body dual body, so you get a " scaled " baseline body - ie a Platonic solids, which can be converted by Centric stretching in the initial article. Both therefore have the same focus.

Symmetry

The Platonic solids show the greatest possible symmetry:

• Corners, edges and faces are mutually similar, that is, every vertex ( edge, face ) can be represented by a symmetry of the body to any other corner ( edge, face ).

One says:

• The symmetry group acts transitively on the corners (as well as on the edges and surfaces ).

It is even true:

• The symmetry group acts transitively on the flags. ( A flag is a corner on a ridge on a surface. )

The five Platonic solids are therefore regular polyhedra. The symmetry groups occurring in them (and their subgroups ) belong to the discrete point groups. Dual Platonic solids have the same symmetry group. This is the basis for the construction of numerous other bodies ( such as the Archimedean bodies). It is thus not five, but only three of these groups: the tetrahedral group, the cube group and the icosahedral group. They play in different contexts in the mathematics involved.

Due to their symmetry made ​​homogeneous models of Platonic bodies have the property that they can fall on each of their faces on a roll with exactly the same probability. Most dice are the way due to the indentations for the eyes figures are not absolutely perfectly symmetrical.

Deltahedra

Since tetrahedron, octahedron and icosahedron belong to the convex deltahedra heard from any symmetry group a body to the deltahedra.

Touching balls

Directly from the high symmetry follows: Each platonic solids has

• An inscribed ball that touches all of its surfaces, and
• A circumscribed sphere, are at all of its corners, and
• An edge sphere on which the center points of the edges.

The common focus of these three spheres is the center (the center ) of the Platonic solid.

Further mathematical properties

Platonic solids as regular tilings of the sphere

If one projects the edges of a Platonic solid from the center point on a sphere with the same center (eg on the circumscribed sphere ), we obtain a tiling of the sphere by mutually congruent regular spherical polygons, in each corner of the same number of edges ( under the same angles ) meet. These tilings have the same symmetries as the original body. In particular, they are also fahnentransitiv. There are the five regular tessellations of the sphere, between which the same duality relations exist as between the bodies. ( In another context one also speaks of maps and dual maps. ) Every regular tiling can be described by the pair, where is the number of edges of a stone and for the number of ending in a corner edges. Therefore, the Platonic solids yield the dual pairs and, and, and the self-dual pair. These are all solutions of the inequality

This relationship follows from the Eulerian Polyedersatz which each represents the number of faces, edges and corners in terms of:

Applies in the plane ( with suitable interpretation, namely asymptotically )

Or

With the solutions

The solutions of

Provide regular tilings of the hyperbolic plane.

From the Platonic solids derived polyhedra

Because of the strong regularity of the Platonic solids can be derived easily other body of them who are very regular again. One has only to apply the same constructions symmetrically on faces, edges or corners. An example of this is the dual body, which result from the fact that it connects the center of each face with the centers of the adjacent land.

Einbeschreibungen

There are certainly other ways to incorporate a Platonic body to another.

For example to give a tetrahedron, when using the diagonal of a cube face as an edge to warped diagonal on the opposing surface than another, and when the other four edges using the diagonal connecting the ends of the two.

An octahedron obtained when one puts surfaces through the centers of the edges of a tetrahedron.

From a cube gives a dodecahedron when setting up a suitable hipped roof on each face; vice versa are obtained by a suitable choice of face diagonals on a dodecahedron, the cube back:

Deadened platonic solids

If you generated from a Platonic body a truncated polyhedron by cutting off its vertices so that after all edges have the same length, we obtain a semi- regular ( Archimedean ) body. This body is also formed as a section of the Platonic solid with his aptly enlarged dual body.

Archimedean solids are examples of fairly regular bodies in which polygons are used, regularly, but are of different page number.

Star body

If you build pyramids on the side surfaces, rather than cut, you get star bodies, such as the star tetrahedron.

If one uses equilateral triangles for the pyramids, one has examples of polyhedra that consist entirely of equal polygons, but in which different numbers of colliding in the corners.

Higher-Dimensional Platonic Solids

The Swiss mathematician Ludwig Schläfli certain 1852 -dimensional relatives of the Platonic solids - but his work remained long unnoticed. It turned out that there is an additional, sixth Platonic solids in four-dimensional space, the 24 - Zeller ( Ikositetrachor ).

In the fifth dimension - and in all later - there are only three instead of five Platonic solids: tetrahedron Hyper, hypercube and Hyperoktaeder.

→ See also: Regular 24 - Zeller ( Ikositetrachor )

History

The Platonic solids have been studied since ancient times. The Pythagoreans ( 6th century BC), a distinction at least between tetrahedron, hexahedron and dodecahedron. Octahedra may not have been noted because it has been considered as a double pyramid. The Theaetetus of Athens ( 415-369 BC) also knew octahedron and icosahedron. He proved that there can be only five convex regular polyhedra.

The Greek philosopher Plato (ca. 427-347 BC), a contemporary of Theaetetus ', the namesake for the five body was. In his work Timaeus ( Ch. 20, 53c4 - 55c6 ) he described it in detail. He tied the Platonic solids in his philosophical system, turning them (except dodecahedron ) the four elements then assigned (Section 21, 55c7 - 56c7 ): Fire stood for the tetrahedron, the octahedron for air. The icosahedron was associated with water, the hexahedron with earth. The dodecahedron could be equated according to this theory postulated by Aristotle with the fifth element ether.

Euclid ( 360-280 BC ) described the Platonic solids in XIII. Book of its elements (§ § 13-17). He demonstrated, among other things, that there are exactly five ( § 18a). Hypsicles took in later appended " XIV Book " before some volume calculations ( from the 2nd century BC). The " XV. Book " (n. from the 6th century BC ) was a Greek mathematician further discoveries regarding the five regular solids.

With the advent of several artists perspective processed the Platonic solids in their works: Piero della Francesca, Leonardo da Vinci ( illustrations to Divina Proportione of Luca Pacioli ), Albrecht Dürer, Wenzel Jamnitzer ( Perspectiva Corporum Regularium, 1568).

Johannes Kepler succeeded ( Mysterium Cosmo, 1596), represent the orbital radii of the six known planets by a certain combination of these five forms and their inner and outer spheres. This interpretation was broadly in line with the then-known astronomical values ​​, but corresponded in fact no law.

Platonic solids beyond the mathematics

The striking regularity makes the Platonic solids in many ways for the people interesting.

• Some Platonic solids are solutions to the problem of Thomson (after Joseph John Thomson ): Clearly spoken describes this problem of how to distribute n electrons on a spherical surface, so that the potential energy is minimal due to their electric field.
• In addition to the classical, geometric cube, which is easy to manufacture and has been used for thousands of years for gambling, find today the other Platonic solids ( also referred to as a cube ) application in the game, such as pen -and- paper role-playing games (see game cube). The prerequisites for this are a physically uniform density distribution - ie homogeneous material - and the uniform nature of all corners and edges.
• Platonic solids are a visual artist long objects. In modern art has dealt mainly MC Escher with them and like regular solids; works by Salvador Dalí thematize platonic solids or their development.
• Platonic polyhedra also play an important role in the adventure game The Dig.
• About the use of the Roman dodecahedron is speculated to date.
• Rudolf von Laban concretized its spatial- rhythmic movement theory ( chorus members ) mainly in the model of the icosahedron.
• In the management of teams could be, according to a proposal by Stafford Beer, using the Platonic solids as a model for cross-linking at concentration of employees on their topics. Each employee corresponding to an edge, each subject a corner of a Platonic solid. For each topic, we meet regularly with exactly the employees, the edges of which converge in this topic area. Example, an employee worked more than two threads at a time and can concentrate well. Even with large teams ( eg icosahedron = 30 people, 5 people per topic, 12 subjects ) was therefore ensures that there is order. Beers idea Gallen was at the Management Centre St and proposes a method based on it called Syntegrity.

Even in nature, existing regularities as platonic solids can mint.

• The arrangement of the hydrogen atoms, for example, in the sp ³ -hybridized methane hybrid orbital corresponds to a tetrahedron.
• Tetrahedron, cube and octahedron occur in nature as ( idealized ) crystals; dodecahedral and icosahedral symmetry elements are found in quasicrystals.
• Exact dodecahedron do not occur as crystals. Crystals of certain minerals, such as pyrite, externally resemble a dodecahedron, are not exact pentagonal dodecahedron, but distorted. However, the distortion to the naked eye from a distance is often not perceived. Up close, you realize, however, that these bodies are not formed from regular ( but irregular ) pentagons. For example, forming sodium chloride and alum, which is doped in the precipitation with certain other materials, cube- crystals. Pure alum crystallizes as the octahedron. The demarcation between the individual forms is not absolute, but the internal symmetry can manifest itself in different forms. In mineralogy, all the Platonic solids, tetrahedron, cube and octahedron and rhombic dodecahedron, cuboctahedron and their mixed forms fall under the cubic term. Many minerals can therefore assume more of these cubic forms. This includes, for example, pyrite, which is present both as a cube or octahedron, and as described above, as a distorted dodecahedron.
• Platonic solids, in particular the icosahedron, are very common structural forms, such as those observed in Clusters ( small nanoparticles).
• Some of the Platonic solids are formed by organic hydrocarbon molecules ( see Platonic hydrocarbons).