In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, a Schlegel diagram is a
projection of a
polytope
In elementary geometry, a polytope is a geometric object with flat sides ('' faces''). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions as an ...
from
into
through a
point just outside one of its
facets. The resulting entity is a
polytopal subdivision of the facet in
that, together with the original facet, is combinatorially equivalent to the original polytope. The diagram is named for
Victor Schlegel, who in 1886 introduced this tool for studying
combinatorial and
topological properties of polytopes. In
dimension
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coord ...
3, a Schlegel diagram is a projection of a
polyhedron
In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices.
A convex polyhedron is the convex hull of finitely many points, not all o ...
into a
plane figure; in
dimension
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coord ...
4, it is a projection of a
4-polytope to
3-space. As such, Schlegel diagrams are commonly used as a means of visualizing
four-dimensional
A four-dimensional space (4D) is a mathematical extension of the concept of three-dimensional or 3D space. Three-dimensional space is the simplest possible abstraction of the observation that one only needs three numbers, called '' dimensions'' ...
polytopes.
Construction
The most elementary Schlegel diagram, that of a polyhedron, was described by
Duncan Sommerville
Duncan MacLaren Young Sommerville (1879–1934) was a Scottish mathematician and astronomer. He compiled a bibliography on non-Euclidean geometry and also wrote a leading textbook in that field. He also wrote ''Introduction to the Geometry of N ...
as follows:
:A very useful method of representing a convex polyhedron is by plane projection. If it is projected from any external point, since each ray cuts it twice, it will be represented by a polygonal area divided twice over into polygons. It is always possible by suitable choice of the centre of projection to make the projection of one face completely contain the projections of all the other faces. This is called a ''Schlegel diagram'' of the polyhedron. The Schlegel diagram completely represents the morphology of the polyhedron. It is sometimes convenient to project the polyhedron from a vertex; this vertex is projected to infinity and does not appear in the diagram, the edges through it are represented by lines drawn outwards.
Sommerville also considers the case of a
simplex
In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. ...
in four dimensions:
[Sommerville (1929), p.101.] "The Schlegel diagram of simplex in S
4 is a
tetrahedron
In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all ...
divided into four tetrahedra." More generally, a polytope in n-dimensions has a Schlegel diagram constructed by a
perspective projection
Linear or point-projection perspective (from la, perspicere 'to see through') is one of two types of graphical projection perspective in the graphic arts; the other is parallel projection. Linear perspective is an approximate representation ...
viewed from a point outside of the polytope, above the center of a facet. All vertices and edges of the polytope are projected onto a
hyperplane
In geometry, a hyperplane is a subspace whose dimension is one less than that of its '' ambient space''. For example, if a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hyper ...
of that facet. If the polytope is convex, a point near the facet will exist which maps the facet outside, and all other facets inside, so no edges need to cross in the projection.
Examples
See also
*
Net (polyhedron)
In geometry, a net of a polyhedron is an arrangement of non-overlapping edge-joined polygons in the plane which can be folded (along edges) to become the faces of the polyhedron. Polyhedral nets are a useful aid to the study of polyhedra and sol ...
– A different approach for visualization by lowering the dimension of a polytope is to build a net, disconnecting facets, and ''unfolding'' until the facets can exist on a single hyperplane. This maintains the geometric scale and shape, but makes the topological connections harder to see.
References
Further reading
*
Victor Schlegel (1883) ''Theorie der homogen zusammengesetzten Raumgebilde'', Nova Acta, Ksl. Leop.-Carol. Deutsche Akademie der Naturforscher, Band XLIV, Nr. 4, Druck von E. Blochmann & Sohn in Dresden
* Victor Schlegel (1886) ''Ueber Projectionsmodelle der regelmässigen vier-dimensionalen Körper'', Waren.
*
Coxeter, Coxeter, H.S.M.; ''
Regular Polytopes'', (Methuen and Co., 1948). (p. 242)
** ''
Regular Polytopes'', (3rd edition, 1973), Dover edition,
* .
External links
*
** {{mathworld , urlname = Skeleton , title = Skeleton
George W. Hart: 4D Polytope Projection Models by 3D PrintingNrich maths – for the teenager. Also useful for teachers.
Polytopes
Projective geometry