In
geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
, a Schlegel diagram is a
projection of a
polytope 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
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and as an end to obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many ...
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 coo ...
3, a Schlegel diagram is a projection of a
polyhedron
In geometry, a polyhedron (: polyhedra or polyhedrons; ) is a three-dimensional figure with flat polygonal Face (geometry), faces, straight Edge (geometry), edges and sharp corners or Vertex (geometry), vertices. The term "polyhedron" may refer ...
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 coo ...
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 polytopes.
Construction
The most elementary Schlegel diagram, that of a polyhedron, was described by
Duncan Sommerville 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 four dimensions:
[Sommerville (1929), p.101.] "The Schlegel diagram of simplex in S
4 is a
tetrahedron
In geometry, a tetrahedron (: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular Face (geometry), faces, six straight Edge (geometry), edges, and four vertex (geometry), vertices. The tet ...
divided into four tetrahedra." More generally, a polytope in n-dimensions has a Schlegel diagram constructed by a
perspective projection 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 generalization of a two-dimensional plane in three-dimensional space to mathematical spaces of arbitrary dimension. Like a plane in space, a hyperplane is a flat hypersurface, a subspace whose dimension is ...
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) – 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