Octahedron
In geometry, an octahedron is a polyhedron with eight faces, twelve edges, six vertices. The term is most 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 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. If the edge length of a regular octahedron is a, the radius of a circumscribed sphere is r u = a 2 2 ≈ 0.707 ⋅ a and the radius of an inscribed sphere is r i = a 6 6 ≈ 0.408 ⋅ a while the midradius, which touches the middle of each edge, is r m = a 2 = 0.5 ⋅ a The octahedron has four special orthogonal projections, centered, on an edge, vertex and normal to a face. The second and third correspond to A2 Coxeter planes.
The octahedron can be represented as a spherical tiling, projected onto the plane via a stereographic projection. This projection is conformal, preserving angles but not lengths. Straight lines on the sphere are projected as circular arcs on the plane. An octahedron with edge length √2 can be placed with its center at the origin and its vertices on the coordinate axes. In an x–y–z Cartesian coordinate system, the octahedron with center coordinates and radius r is the set of all points such that | x − a | + | y − b | + | z − c | = r; the surface area A and the volume V of a regular octahedron of edge length a are: A = 2 3 a 2 ≈ 3.464 a 2 V = 1 3 2 a 3 ≈ 0.471 a 3 Thus the volume is four times that of a regular tetrahedron with the same edge length, while the surface area is twice. If an octahedron has been stretched so that it obeys the equation | x x m | + | y y m | + | z z m | = 1, the formulas for the surface area and volume expand to become A = 4 x m y m z m × 1 x m 2 + 1 y m 2 + 1 z m 2, V = 4 3 x m y m z m.
Additionally the inertia tensor of the stretched octahedron is I =. These reduce to the equations for the regular octahedron when x m = y m = z m = a 2 2. {\displaystyle x_=y_=z_=
Wythoff symbol
In geometry, the Wythoff symbol represents a Wythoff construction of a uniform polyhedron or plane tiling, from a Schwarz triangle. It was first used by Coxeter, Longuet-Higgins and Miller in their enumeration of the uniform polyhedra. A Wythoff symbol consists of a vertical bar, it represents one uniform polyhedron or tiling, although the same tiling/polyhedron can have different Wythoff symbols from different symmetry generators. For example, the regular cube can be represented by 3 | 4 2 with Oh symmetry, 2 4 | 2 as a square prism with 2 colors and D4h symmetry, as well as 2 2 2 | with 3 colors and D 2 h symmetry. With a slight extension, Wythoff's symbol can be applied to all uniform polyhedra. However, the construction methods do not lead to all uniform tilings in Euclidean or hyperbolic space. In three dimensions, Wythoff's construction begins by choosing a generator point on the triangle. If the distance of this point from each of the sides is non-zero, the point must be chosen to be an equal distance from each edge.
A perpendicular line is dropped between the generator point and every face that it does not lie on. The three numbers in Wythoff's symbol, p, q and r, represent the corners of the Schwarz triangle used in the construction, which are π / p, π / q and π / r radians respectively; the triangle is represented with the same numbers, written. The vertical bar in the symbol specifies a categorical position of the generator point within the fundamental triangle according to the following: p | q r indicates that the generator lies on the corner p, p q | r indicates that the generator lies on the edge between p and q, p q r | indicates that the generator lies in the interior of the triangle. In this notation the mirrors are labeled by the reflection-order of the opposite vertex; the p, q, r values are listed before the bar. The one impossible symbol | p q r implies the generator point is on all mirrors, only possible if the triangle is degenerate, reduced to a point; this unused symbol is therefore arbitrarily reassigned to represent the case where all mirrors are active, but odd-numbered reflected images are ignored.
The resulting figure has rotational symmetry only. The generator point can either be off each mirror, activated or not; this distinction creates 8 possible forms, neglecting one where the generator point is on all the mirrors. The Wythoff symbol is functionally similar to the more general Coxeter-Dynkin diagram, in which each node represents a mirror and the arcs between them – marked with numbers – the angles between the mirrors. A node is circled. There are seven generator points with each set of p, q, r: There are three special cases: p q | – This is a mixture of p q r | and p q s |, containing only the faces shared by both. | p q r – Snub forms are given by this otherwise unused symbol. | p q r s – A unique snub form for U75 that isn't Wythoff-constructible. There are 4 symmetry classes of reflection on the sphere, three in the Euclidean plane. A few of the infinitely many such patterns in the hyperbolic plane are listed. Point groups: dihedral symmetry, p = 2, 3, 4 … tetrahedral symmetry octahedral symmetry icosahedral symmetry Euclidean groups: *442 symmetry: 45°-45°-90° triangle *632 symmetry: 30°-60°-90° triangle *333 symmetry: 60°-60°-60° triangleHyperbolic groups: *732 symmetry *832 symmetry *433 symmetry *443 symmetry *444 symmetry *542 symmetry *642 symmetry...
The above symmetry groups only include the integer solutions on the sphere. The list of Schwarz triangles includes rational numbers, determine the full set of solutions of nonconvex uniform polyhedra. In the tilings above, each triangle is a fundamental domain, colored by and odd reflections. Selected tilings created by the Wythoff con
Hyperbolic geometry
In mathematics, hyperbolic geometry is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: For any given line R and point P not on R, in the plane containing both line R and point P there are at least two distinct lines through P that do not intersect R. Hyperbolic plane geometry is the geometry of saddle surfaces and pseudospherical surfaces, surfaces with a constant negative Gaussian curvature. A modern use of hyperbolic geometry is in the theory of special relativity Minkowski spacetime and gyrovector space; when geometers first realised they were working with something other than the standard Euclidean geometry they described their geometry under many different names. In the former Soviet Union, it is called Lobachevskian geometry, named after one of its discoverers, the Russian geometer Nikolai Lobachevsky; this page is about the 2-dimensional hyperbolic geometry and the differences and similarities between Euclidean and hyperbolic geometry. Hyperbolic geometry can be extended to three and more dimensions.
Hyperbolic geometry is more related to Euclidean geometry than it seems: the only axiomatic difference is the parallel postulate. When the parallel postulate is removed from Euclidean geometry the resulting geometry is absolute geometry. There are two kinds of absolute geometry and hyperbolic. All theorems of absolute geometry, including the first 28 propositions of book one of Euclid's Elements, are valid in Euclidean and hyperbolic geometry. Propositions 27 and 28 of Book One of Euclid's Elements prove the existence of parallel/non-intersecting lines; this difference has many consequences: concepts that are equivalent in Euclidean geometry are not equivalent in hyperbolic geometry. Further, because of the angle of parallelism, hyperbolic geometry has an absolute scale, a relation between distance and angle measurements. Single lines in hyperbolic geometry have the same properties as single straight lines in Euclidean geometry. For example, two points uniquely define a line, lines can be infinitely extended.
Two intersecting lines have the same properties as two intersecting lines in Euclidean geometry. For example, two lines can intersect in no more than one point, intersecting lines have equal opposite angles, adjacent angles of intersecting lines are supplementary; when we add a third line there are properties of intersecting lines that differ from intersecting lines in Euclidean geometry. For example, given 2 intersecting lines there are infinitely many lines that do not intersect either of the given lines; these properties all are independent of the model used if the lines may look radically different. Non-intersecting lines in hyperbolic geometry have properties that differ from non-intersecting lines in Euclidean geometry: For any line R and any point P which does not lie on R, in the plane containing line R and point P there are at least two distinct lines through P that do not intersect R; this implies that there are through P an infinite number of coplanar lines that do not intersect R.
These non-intersecting lines are divided into two classes: Two of the lines are limiting parallels: there is one in the direction of each of the ideal points at the "ends" of R, asymptotically approaching R, always getting closer to R, but never meeting it. All other non-intersecting lines have a point of minimum distance and diverge from both sides of that point, are called ultraparallel, diverging parallel or sometimes non-intersecting; some geometers use parallel lines instead of limiting parallel lines, with ultraparallel lines being just non-intersecting. These limiting parallels make an angle θ with PB. For ultraparallel lines, the ultraparallel theorem states that there is a unique line in the hyperbolic plane, perpendicular to each pair of ultraparallel lines. In hyperbolic geometry, the circumference of a circle of radius r is greater than 2 π r. Let R = 1 − K, where K is the Gaussian curvature of the plane. In hyperbolic geometry, K is negative, so the square root is of a positive number.
The circumference of a circle of radius r is equal to: 2 π R sinh r R. And the area of the enclosed disk is: 4 π R 2 sinh 2 r 2 R = 2 π R 2. Therefore, in hyperbolic geometry the ratio of a circle's circumference to its radius is always greater than 2 π, though
Isotoxal figure
In geometry, a polytope, or a tiling, is isotoxal or edge-transitive if its symmetries act transitively on its edges. Informally, this means that there is only one type of edge to the object: given two edges, there is a translation, rotation and/or reflection that will move one edge to the other, while leaving the region occupied by the object unchanged; the term isotoxal is derived from the Greek τοξον meaning arc. An isotoxal polygon is an equilateral polygon; the duals of isotoxal polygons are isogonal polygons. In general, an isotoxal 2n-gon will have Dn dihedral symmetry. A rhombus is an isotoxal polygon with D2 symmetry. All regular polygons are isotoxal, having double the minimum symmetry order: a regular n-gon has Dn dihedral symmetry. A regular 2n-gon is an isotoxal polygon and can be marked with alternately colored vertices, removing the line of reflection through the mid-edges. Regular polyhedra are isohedral and isotoxal. Quasiregular polyhedra are not isohedral. Not every polyhedron or 2-dimensional tessellation constructed from regular polygons is isotoxal.
For instance, the truncated icosahedron has two types of edges: hexagon-hexagon and hexagon-pentagon, it is not possible for a symmetry of the solid to move a hexagon-hexagon edge onto a hexagon-pentagon edge. An isotoxal polyhedron has the same dihedral angle for all edges. There are nine convex isotoxal polyhedra formed from the Platonic solids, 8 formed by the Kepler–Poinsot polyhedra, six more as quasiregular star polyhedra and their duals. There are at least 5 polygonal tilings of the Euclidean plane that are isotoxal, infinitely many isotoxal polygonal tilings of the hyperbolic plane, including the Wythoff constructions from the regular hyperbolic tilings, non-right groups. Table of polyhedron dihedral angles Vertex-transitive Face-transitive Cell-transitive Peter R. Cromwell, Cambridge University Press 1997, ISBN 0-521-55432-2, p. 371 Transitivity Grünbaum, Branko. C.. Tilings and Patterns. New York: W. H. Freeman. ISBN 0-7167-1193-1. CS1 maint: Multiple names: authors list Coxeter, Harold Scott MacDonald.
"Uniform polyhedra", Philosophical Transactions of the Royal Society of London. Series A. Mathematical and Physical Sciences, 246: 401–450, doi:10.1098/rsta.1954.0003, ISSN 0080-4614, JSTOR 91532, MR 0062446
Octagon
In geometry, an octagon is an eight-sided polygon or 8-gon. A regular octagon has Schläfli symbol and can be constructed as a quasiregular truncated square, t, which alternates two types of edges. A truncated octagon, t; the sum of all the internal angles of any octagon is 1080°. As with all polygons, the external angles total 360°. If squares are constructed all internally or all externally on the sides of an octagon the midpoints of the segments connecting the centers of opposite squares form a quadrilateral, both equidiagonal and orthodiagonal; the midpoint octagon of a reference octagon has its eight vertices at the midpoints of the sides of the reference octagon. If squares are constructed all internally or all externally on the sides of the midpoint octagon the midpoints of the segments connecting the centers of opposite squares themselves form the vertices of a square. A regular octagon is a closed figure with sides of the same length and internal angles of the same size, it has eight lines of reflective symmetry and rotational symmetry of order 8.
A regular octagon is represented by the Schläfli symbol. The internal angle at each vertex of a regular octagon is 135°; the central angle is 45°. The area of a regular octagon of side length a is given by A = 2 cot π 8 a 2 = 2 a 2 ≃ 4.828 a 2. In terms of the circumradius R, the area is A = 4 sin π 4 R 2 = 2 2 R 2 ≃ 2.828 R 2. In terms of the apothem r, the area is A = 8 tan π 8 r 2 = 8 r 2 ≃ 3.314 r 2. These last two coefficients bracket the value of the area of the unit circle; the area can be expressed as A = S 2 − a 2, where S is the span of the octagon, or the second-shortest diagonal. This is proven if one takes an octagon, draws a square around the outside and takes the corner triangles and places them with right angles pointed inward, forming a square; the edges of this square are each the length of the base. Given the length of a side a, the span S is S = a 2 + a + a 2 = a ≈ 2.414 a. The area is as above: A = 2 − a 2 = 2 a 2 ≈ 4.828 a 2. Expressed in terms of the span, the area is A = 2 S 2 ≈ 0.828 S 2.
Another simple formula for the area is A = 2 a S. More the span S is known, the length of the sides, a, is to be determined, as when cutting a square piece of material into a regular octagon. From the above, a ≈ S / 2.414. The two end lengths e on each side, as well as being e = a / 2, may be calculated as e = / 2; the circumradius of the regular octagon in terms of the side length a is R = a, the inradius is r = a. The regular octagon, in ter
Checkerboard
A checkerboard or chequerboard is a board of chequered pattern on which draughts is played. Most it consists of 64 squares of alternating dark and light color green and buff and red, or black and white. An 8×8 checkerboard is used to play many other games, including chess, whereby it is known as a chessboard. Other rectangular square-tiled boards are often called checkerboards. Given a matrix with m rows and n columns, a function f, f = { black if m ∧ 1 = n ∧ 1, white if m ∧ 1 ≠ n ∧ 1 or, alternatively, f = { black if m + n is white if m + n is odd The element = is black and represents the lower left corner of the board. Martin Gardner featured puzzles based on checkerboards in his November 1962 Mathematical Games column in Scientific American. A square checkerboard with an alternating pattern is used for games including: Amazons Chapayev Chess and some of its variants Czech draughts Draughts known as checkers Frisian draughts Gounki International draughts Italian draughts Lines of Action Pool checkers Russian checkersThe following games require an 8×8 board and are sometimes played on a chessboard.
Arimaa Breakthrough Crossings Mak-yek Makruk Martian Chess
Kaleidoscope
A kaleidoscope is an optical instrument with two or more reflecting surfaces tilted to each other in an angle, so that one or more objects on one end of the mirrors are seen as a regular symmetrical pattern when viewed from the other end, due to repeated reflection. The reflectors are enclosed in a tube containing on one end a cell with loose, colored pieces of glass or other transparent materials to be reflected into the viewed pattern. Rotation of the cell causes motion of the materials, resulting in an ever-changing view being presented. Coined by its Scottish inventor David Brewster, "kaleidoscope" is derived from the Ancient Greek word καλός, "beautiful, beauty", εἶδος, "that, seen: form, shape" and σκοπέω, "to look to, to examine", hence "observation of beautiful forms." It was first published in the patent, granted on July 10, 1817. Multiple reflection by two or more reflecting surfaces has been known since antiquity and was described as such by Giambattista della Porta in his Magia Naturalis.
In 1646 Athanasius Kircher described an experiment with a construction of two mirrors, which could be opened and closed like a book and positioned in various angles, showing regular polygon figures consisting of reflected aliquot sectors of 360°. Mr. Bradley's New Improvements in Planting and Gardening described a similar construction to be placed on geometrical drawings to show an image with multiplied reflection. However, an optimal configuration that produces the full effects of the kaleidoscope was not recorded before 1815. In 1814 Sir David Brewster conducted experiments on light polarization by successive reflections between plates of glass and first noted "the circular arrangement of the images of a candle round a center, the multiplication of the sectors formed by the extremities of the plates of glass", he forgot about it, but noticed a more impressive version of the effect during further experiments in February 1815. A while he was impressed by the multiplied reflection of a bit of cement, pressed through at the end of a triangular glass trough, which appeared more regular and perfectly symmetrical in comparison to the reflected objects, situated further away from the reflecting plates in earlier experiments.
This triggered more experiments to find the conditions for the most beautiful and symmetrically perfect conditions. An early version had pieces of colored glass and other irregular objects fixed permanently and was admired by some Members of the Royal Society of Edinburgh, including Sir George Mackenzie who predicted its popularity. A version followed in which some of the objects and pieces of glass could move when the tube was rotated; the last step, regarded as most important by Brewster, was to place the reflecting panes in a draw tube with a concave lens to distinctly introduce surrounding objects into the reflected pattern. Brewster thought his instrument to be of great value in "all the ornamental arts" as a device that creates an "infinity of patterns". Artists could delineate the produced figures of the kaleidoscope by means of the solar microscope, magic lantern or camera lucida. Brewster believed it would at the same time become a popular instrument "for the purposes of rational amusement".
He decided to apply for a patent. British patent no. 4136 "for a new Optical Instrument called "The Kaleidoscope" for exhibiting and creating beautiful Forms and Patterns of great use in all the ornamental Arts" was granted in July 1817. The manufacturer engaged to produce the product had shown one of the patent instruments to some of the London opticians to see if he could get orders from them. Soon the instrument was copied and marketed before the manufacturer had prepared any number of kaleidoscopes for sale. An estimated two hundred thousand kaleidoscopes sold in London and Paris in just three months. Brewster figured at most a thousand of these were authorized copies that were constructed while the majority of the others did not give a correct impression of his invention; because so few people had experienced a proper kaleidoscope or knew how to apply it to ornamental arts, he decided to publicize a treatise on the principles and the correct construction of the kaleidoscope. It was thought that the patent was reduced in a Court of Law since its principles were already known.
Brewster stated that the kaleidoscope was different because the particular positions of the object and of the eye, played a important role in producing the beautiful symmetrical forms. Brewster's opinion was shared including James Watt. Philip Carpenter tried to produce his own imitation of the kaleidoscope, but was not satisfied with the results, he decided to offer his services to Brewster as manufacturer. Brewster agreed and Carpenter's models were stamped "sole maker". Realizing that the company could not meet the level of demand, Brewster gained permission from Carpenter in 1818 for the device to be made by other manufacturers. In his 1819 Treatise on the Kaleidoscope Brewster listed more than a dozen manufacturers/sellers of patent kaleidoscopes. Carpenter's company would keep on selling kaleidoscopes for 60 years. H. M. Quackenbush Co. based in upstate New York in the United States was another authorized manufacturer. In 1987, kaleidoscope artist Thea Marshall, working with the Willamette Science and Technology Center, a science museum located in the Eugene, Oregon and constructed a 1,000 square foot traveling mathematics and science exhibition, "Kaleidoscopes: Reflections of Science and Art."
With funding from the National Science Foundation, circulated under the auspices of the Smithsonian Insti