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In geometry, a convex polyhedron whose faces are regular polygons is known as a Johnson solid, or sometimes as a Johnson–Zalgaller solid [1].Some authors exclude uniform polyhedra (in which all vertices are symmetric to each other) from the definition; uniform polyhedra include Platonic and Archimedean solids as well as prisms and antiprisms. [2]
The "nine dots" puzzle. The puzzle asks to link all nine dots using four straight lines or fewer, without lifting the pen. The nine dots puzzle is a mathematical puzzle whose task is to connect nine squarely arranged points with a pen by four (or fewer) straight lines without lifting the pen or retracing any lines.
In mathematics, solid partitions are natural generalizations of integer partitions and plane partitions defined by Percy Alexander MacMahon. [1] A solid partition of n {\displaystyle n} is a three-dimensional array of non-negative integers n i , j , k {\displaystyle n_{i,j,k}} (with indices i , j , k ≥ 1 {\displaystyle i,j,k\geq 1} ) such that
A Johnson solid is a convex polyhedron whose faces are all regular polygons. [2] The convex polyhedron means as bounded intersections of finitely many half-spaces, or as the convex hull of finitely many points. [3]
These semiregular solids can be fully specified by a vertex configuration: a listing of the faces by number of sides, in order as they occur around a vertex. For example: 3.5.3.5 represents the icosidodecahedron, which alternates two triangles and two pentagons around each vertex. In contrast: 3.3.3.5 is a pentagonal antiprism.
Coxeter, Longuet-Higgins & Miller (1954) define uniform polyhedra to be vertex-transitive polyhedra with regular faces. They define a polyhedron to be a finite set of polygons such that each side of a polygon is a side of just one other polygon, such that no non-empty proper subset of the polygons has the same property.
A sphere of radius r has surface area 4πr 2.. The surface area (symbol A) of a solid object is a measure of the total area that the surface of the object occupies. [1] The mathematical definition of surface area in the presence of curved surfaces is considerably more involved than the definition of arc length of one-dimensional curves, or of the surface area for polyhedra (i.e., objects with ...
group [6] Truncated tetrahedron: 3.6.6: 4 triangles 4 hexagons: 18 12 T d: Cuboctahedron: 3.4.3.4: 8 triangles 6 squares: 24 12 O h: Truncated cube: 3.8.8: 8 triangles 6 octagons: 36 24 O h: Truncated octahedron: 4.6.6: 6 squares 8 hexagons 36 24 O h: Rhombicuboctahedron: 3.4.4.4: 8 triangles 18 squares 48 24 O h: Truncated cuboctahedron: 4.6.8 ...
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