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Prince Rupert's Cube
In geometry, Prince Rupert's cube is the largest cube that can pass through a hole cut through a unit cube without splitting it into two pieces. Its side length is approximately 1.06, 6% larger than the side length 1 of the unit cube through which it passes. The problem of finding the largest square that lies entirely within a unit cube is closely related, and has the same solution. Prince Rupert's cube is named after Prince Rupert of the Rhine, who asked whether a cube could be passed through a hole made in another cube ''of the same size'' without splitting the cube into two pieces. A positive answer was given by John Wallis. Approximately 100 years later, Pieter Nieuwland found the largest possible cube that can pass through a hole in a unit cube. Many other convex polyhedra, including all five Platonic solids, have been shown to have the ''Rupert property'': a copy of the polyhedron, of the same or larger shape, can be passed through a hole in the polyhedron. It is unkn ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prince Ruperts Cube
A prince is a Monarch, male ruler (ranked below a king, grand prince, and grand duke) or a male member of a monarch's or former monarch's family. ''Prince'' is also a title of nobility (often highest), often hereditary title, hereditary, in some European State (polity), states. The female equivalent is a princess. The English language, English word derives, via the French language, French word ''prince'', from the Latin noun , from (first) and (head), meaning "the first, foremost, the chief, most distinguished, noble monarch, ruler, prince". Historical background The Latin word (older Latin *prīsmo-kaps, literally "the one who takes the first [place/position]"), became the usual title of the informal leader of the Roman senate some centuries before the transition to Roman Empire, empire, the ''princeps senatus''. Emperor Augustus established the formal position of monarch on the basis of principate, not Dominate, dominion. He also tasked his grandsons as summer rulers o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regular Hexagon
In geometry, a hexagon (from Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°. Regular hexagon A '' regular hexagon'' has Schläfli symbol and can also be constructed as a truncated equilateral triangle, t, which alternates two types of edges. A regular hexagon is defined as a hexagon that is both equilateral and equiangular. It is bicentric, meaning that it is both cyclic (has a circumscribed circle) and tangential (has an inscribed circle). The common length of the sides equals the radius of the circumscribed circle or circumcircle, which equals \tfrac times the apothem (radius of the inscribed circle). All internal angles are 120 degrees. A regular hexagon has six rotational symmetries (''rotational symmetry of order six'') and six reflection symmetries (''six lines of symmetry''), making up the dihedral group D6. The longest diagonals of a regular ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rhombicuboctahedron
In geometry, the rhombicuboctahedron, or small rhombicuboctahedron, is a polyhedron with eight triangular, six square, and twelve rectangular faces. There are 24 identical vertices, with one triangle, one square, and two rectangles meeting at each one. If all the rectangles are themselves square (equivalently, all the edges are the same length, ensuring the triangles are equilateral), it is an Archimedean solid. The polyhedron has octahedral symmetry, like the cube and octahedron. Its dual is called the deltoidal icositetrahedron or trapezoidal icositetrahedron, although its faces are not really true trapezoids. Names Johannes Kepler in Harmonices Mundi (1618) named this polyhedron a ''rhombicuboctahedron'', being short for ''truncated cuboctahedral rhombus'', with ''cuboctahedral rhombus'' being his name for a rhombic dodecahedron. There are different truncations of a rhombic dodecahedron into a topological rhombicuboctahedron: Prominently its rectification (left), the one t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Truncated Cube
In geometry, the truncated cube, or truncated hexahedron, is an Archimedean solid. It has 14 regular faces (6 octagonal and 8 triangular), 36 edges, and 24 vertices. If the truncated cube has unit edge length, its dual triakis octahedron has edges of lengths 2 and 2 + . Area and volume The area ''A'' and the volume ''V'' of a truncated cube of edge length ''a'' are: :\begin A &= 2\left(6+6\sqrt+\sqrt\right)a^2 &&\approx 32.434\,6644a^2 \\ V &= \fraca^3 &&\approx 13.599\,6633a^3. \end Orthogonal projections The ''truncated cube'' has five special orthogonal projections, centered, on a vertex, on two types of edges, and two types of faces: triangles, and octagons. The last two correspond to the B2 and A2 Coxeter planes. Spherical tiling The truncated cube can also be represented as a spherical tiling, and projected onto the plane via a stereographic projection. This projection is conformal, preserving angles but not areas or lengths. Straight lines on the sphere are pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Truncated Octahedron
In geometry, the truncated octahedron is the Archimedean solid that arises from a regular octahedron by removing six pyramids, one at each of the octahedron's vertices. The truncated octahedron has 14 faces (8 regular hexagon, hexagons and 6 Square (geometry), squares), 36 edges, and 24 vertices. Since each of its faces has point symmetry the truncated octahedron is a 6-zonohedron. It is also the Goldberg polyhedron GIV(1,1), containing square and hexagonal faces. Like the cube, it can tessellate (or "pack") 3-dimensional space, as a permutohedron. The truncated octahedron was called the "mecon" by Buckminster Fuller. Its dual polyhedron is the tetrakis hexahedron. If the original truncated octahedron has unit edge length, its dual tetrakis hexahedron has edge lengths and . Construction A truncated octahedron is constructed from a regular octahedron with side length 3''a'' by the removal of six right square pyramids, one from each point. These pyramids have both base side len ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cuboctahedron
A cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertices, with 2 triangles and 2 squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such, it is a quasiregular polyhedron, i.e. an Archimedean solid that is not only vertex-transitive but also edge-transitive. It is radially equilateral. Its dual polyhedron is the rhombic dodecahedron. The cuboctahedron was probably known to Plato: Heron's ''Definitiones'' quotes Archimedes as saying that Plato knew of a solid made of 8 triangles and 6 squares. Synonyms *''Vector Equilibrium'' (Buckminster Fuller) because its center-to-vertex radius equals its edge length (it has radial equilateral symmetry). Fuller also called a cuboctahedron built of rigid struts and flexible vertices a ''jitterbug''; this object can be progressively transformed into an icosahedron, octahedron, and tetrahedron by folding along the diagonals of its square sid ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Archimedean Solid
In geometry, an Archimedean solid is one of the 13 solids first enumerated by Archimedes. They are the convex uniform polyhedra composed of regular polygons meeting in identical vertices, excluding the five Platonic solids (which are composed of only one type of polygon), excluding the prisms and antiprisms, and excluding the pseudorhombicuboctahedron. They are a subset of the Johnson solids, whose regular polygonal faces do not need to meet in identical vertices. "Identical vertices" means that each two vertices are symmetric to each other: A global isometry of the entire solid takes one vertex to the other while laying the solid directly on its initial position. observed that a 14th polyhedron, the elongated square gyrobicupola (or pseudo-rhombicuboctahedron), meets a weaker definition of an Archimedean solid, in which "identical vertices" means merely that the faces surrounding each vertex are of the same types (i.e. each vertex looks the same from close up), so only a lo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Icosahedron
In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes and . The plural can be either "icosahedra" () or "icosahedrons". There are infinitely many non- similar shapes of icosahedra, some of them being more symmetrical than others. The best known is the (convex, non- stellated) regular icosahedron—one of the Platonic solids—whose faces are 20 equilateral triangles. Regular icosahedra There are two objects, one convex and one nonconvex, that can both be called regular icosahedra. Each has 30 edges and 20 equilateral triangle faces with five meeting at each of its twelve vertices. Both have icosahedral symmetry. The term "regular icosahedron" generally refers to the convex variety, while the nonconvex form is called a ''great icosahedron''. Convex regular icosahedron The convex regular icosahedron is usually referred to simply as the ''regular icosahedron'', one of the five regular Platonic solids, and is represented by its Schläfli symbol , con ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dodecahedron
In geometry, a dodecahedron (Greek , from ''dōdeka'' "twelve" + ''hédra'' "base", "seat" or "face") or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagons as faces, which is a Platonic solid. There are also three regular star dodecahedra, which are constructed as stellations of the convex form. All of these have icosahedral symmetry, order 120. Some dodecahedra have the same combinatorial structure as the regular dodecahedron (in terms of the graph formed by its vertices and edges), but their pentagonal faces are not regular: The pyritohedron, a common crystal form in pyrite, has pyritohedral symmetry, while the tetartoid has tetrahedral symmetry. The rhombic dodecahedron can be seen as a limiting case of the pyritohedron, and it has octahedral symmetry. The elongated dodecahedron and trapezo-rhombic dodecahedron variations, along with the rhombic dodecahedra, are space-filling. There ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Octahedron
In geometry, an octahedron (plural: octahedra, octahedrons) is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex. A regular octahedron is the dual polyhedron of a cube. It is a rectified tetrahedron. It is a square bipyramid in any of three orthogonal orientations. It is also a triangular antiprism in any of four orientations. An octahedron is the three-dimensional case of the more general concept of a cross polytope. A regular octahedron is a 3-ball in the Manhattan () metric. Regular octahedron Dimensions If the edge length of a regular octahedron is ''a'', the radius of a circumscribed sphere (one that touches the octahedron at all vertices) is :r_u = \frac a \approx 0.707 \cdot a and the radius of an inscribed sphere (tangent to each of the octahedron's faces) is :r_i = \frac a \approx 0.408\cdot a while the midradius, which ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
3D Printing
3D printing or additive manufacturing is the Manufacturing, construction of a three-dimensional object from a computer-aided design, CAD model or a digital 3D modeling, 3D model. It can be done in a variety of processes in which material is deposited, joined or solidified under Computer Numerical Control, computer control, with material being added together (such as plastics, liquids or powder grains being fused), typically layer by layer. In the 1980s, 3D printing techniques were considered suitable only for the production of functional or aesthetic prototypes, and a more appropriate term for it at the time was rapid prototyping. , the precision, repeatability, and material range of 3D printing have increased to the point that some 3D printing processes are considered viable as an industrial-production technology, whereby the term ''additive manufacturing'' can be used synonymously with ''3D printing''. One of the key advantages of 3D printing is the ability to produce very ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
3D-printed Prince Rupert Cube
3D printing or additive manufacturing is the construction of a three-dimensional object from a CAD model or a digital 3D model. It can be done in a variety of processes in which material is deposited, joined or solidified under computer control, with material being added together (such as plastics, liquids or powder grains being fused), typically layer by layer. In the 1980s, 3D printing techniques were considered suitable only for the production of functional or aesthetic prototypes, and a more appropriate term for it at the time was rapid prototyping. , the precision, repeatability, and material range of 3D printing have increased to the point that some 3D printing processes are considered viable as an industrial-production technology, whereby the term ''additive manufacturing'' can be used synonymously with ''3D printing''. One of the key advantages of 3D printing is the ability to produce very complex shapes or geometries that would be otherwise impossible to construct ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
