Hexadecagon
In mathematics, a hexadecagon is a sixteen-sided polygon. A regular hexadecagon is a hexadecagon in which all angles are equal and all sides are congruent, its Schläfli symbol is and can be constructed as a truncated octagon, t, a twice-truncated square tt. A truncated hexadecagon, t, is a triacontadigon; as 16 = 24, a regular hexadecagon is constructible using compass and straightedge: this was known to ancient Greek mathematicians. Each angle of a regular hexadecagon is 157.5 degrees, the total angle measure of any hexadecagon is 2520 degrees. The area of a regular hexadecagon with edge length t is A = 4 t 2 cot π 16 = 4 t 2 = 4 t 2; because the hexadecagon has a number of sides, a power of two, its area can be computed in terms of the circumradius R by truncating Viète's formula: A = R 2 ⋅ 2 1 ⋅ 2 2 ⋅ 2 2 + 2 = 4 R 2 2 − 2. Since the area of the circumcircle is π R 2, the regular hexadecagon fills 97.45% of its circumcircle. The regular hexadecagon has Dih16 symmetry, order 32. There are 4 dihedral subgroups: Dih8, Dih4, Dih2, Dih1, 5 cyclic subgroups: Z16, Z8, Z4, Z2, Z1, the last implying no symmetry.
On the regular hexadecagon, there are 14 distinct symmetries. John Conway labels full symmetry as r32 and no symmetry is labeled a1; the dihedral symmetries are divided depending on whether they pass through vertices or edges Cyclic symmetries in the middle column are labeled as g for their central gyration orders. The most common high symmetry hexadecagons are d16, a isogonal hexadecagon constructed by eight mirrors can alternate long and short edges, p16, an isotoxal hexadecagon constructed with equal edge lengths, but vertices alternating two different internal angles; these two forms have half the symmetry order of the regular hexadecagon. Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the g16 subgroup has no degrees of freedom. Coxeter states. In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. For the regular hexadecagon, m=8, it can be divided into 28: 4 squares and 3 sets of 8 rhombs.
This decomposition is based with 28 of 1792 faces. The list OEIS: A006245 enumerates the number of solutions as 1232944, including up to 16-fold rotations and chiral forms in reflection. A skew hexadecagon is a skew polygon with edges but not existing on the same plane; the interior of such an hexadecagon is not defined. A skew zig-zag. A regular skew hexadecagon is vertex-transitive with equal edge lengths. In 3-dimensions it will be a zig-zag skew hexadecagon and can be seen in the vertices and side edges of a octagonal antiprism with the same D8d, order 32; the octagrammic antiprism, s and octagrammic crossed-antiprism, s have regular skew octagons. The regular hexadecagon is the Petrie polygon for many higher-dimensional polytopes, shown in these skew orthogonal projections, including: A hexadecagram is a 16-sided star polygon, represented by symbol. There are three regular star polygons, using the same vertices, but connecting every third, fifth or seventh points. There are three compounds: is reduced to 2 as two octagons, is reduced to 4 as four squares and reduces to 2 as two octagrams, is reduced to 8 as eight digons.
Deeper truncations of the regular octagon and octagram can produce isogonal intermediate hexadecagram forms with spaced vertices and two edge lengths. A truncated octagon is a hexadecagon, t=. A quasitruncated octagon, inverted as, is a hexadecagram: t=. A truncated octagram is a hexadecagram: t= and a quasitruncated octagram, inverted as, is a hexadecagram: t=. In the early 16th century, Raphael was the first to construct a perspective image of a regular hexadecagon: the tower in his painting The Ma
Dual polyhedron
In geometry, any polyhedron is associated with a second dual figure, where the vertices of one correspond to the faces of the other and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other. Such dual figures remain combinatorial or abstract polyhedra, but not all are geometric polyhedra. Starting with any given polyhedron, the dual of its dual is the original polyhedron. Duality preserves the symmetries of a polyhedron. Therefore, for many classes of polyhedra defined by their symmetries, the duals belong to a symmetric class. Thus, the regular polyhedra – the Platonic solids and Kepler–Poinsot polyhedra – form dual pairs, where the regular tetrahedron is self-dual; the dual of an isogonal polyhedron, having equivalent vertices, is one, isohedral, having equivalent faces. The dual of an isotoxal polyhedron is isotoxal. Duality is related to reciprocity or polarity, a geometric transformation that, when applied to a convex polyhedron, realizes the dual polyhedron as another convex polyhedron.
There are many kinds of duality. The kinds most relevant to elementary polyhedra are polar reciprocity and topological or abstract duality; the duality of polyhedra is defined in terms of polar reciprocation about a concentric sphere. Here, each vertex is associated with a face plane so that the ray from the center to the vertex is perpendicular to the plane, the product of the distances from the center to each is equal to the square of the radius. In coordinates, for reciprocation about the sphere x 2 + y 2 + z 2 = r 2, the vertex is associated with the plane x 0 x + y 0 y + z 0 z = r 2; the vertices of the dual are the poles reciprocal to the face planes of the original, the faces of the dual lie in the polars reciprocal to the vertices of the original. Any two adjacent vertices define an edge, these will reciprocate to two adjacent faces which intersect to define an edge of the dual; this dual pair of edges are always orthogonal to each other. If r 0 is the radius of the sphere, r 1 and r 2 the distances from its centre to the pole and its polar, then: r 1.
R 2 = r 0 2 For the more symmetrical polyhedra having an obvious centroid, it is common to make the polyhedron and sphere concentric, as in the Dorman Luke construction described below. However, it is possible to reciprocate a polyhedron about any sphere, the resulting form of the dual will depend on the size and position of the sphere; the choice of center for the sphere is sufficient to define the dual up to similarity. If multiple symmetry axes are present, they will intersect at a single point, this is taken to be the centroid. Failing that, a circumscribed sphere, inscribed sphere, or midsphere is used. If a polyhedron in Euclidean space has an element passing through the center of the sphere, the corresponding element of its dual will go to infinity. Since Euclidean space never reaches infinity, the projective equivalent, called extended Euclidean space, may be formed by adding the required'plane at infinity'; some theorists prefer to say that there is no dual. Meanwhile, Wenninger found a way to represent these infinite duals, in a manner suitable for making models.
The concept of duality here is related to the duality in projective geometry, where lines and edges are interchanged. Projective polarity works well enough for convex polyhedra, but for non-convex figures such as star polyhedra, when we seek to rigorously define this form of polyhedral duality in terms of projective polarity, various problems appear. Because of the definitional issues for geometric duality of non-convex polyhedra, Grünbaum argues that any proper definition of a non-convex polyhedron should include a notion of a dual polyhedron. Any convex polyhedron can be distorted into a canonical form, in which a unit midsphere exists tangent to every edge, such that the average position of the points of tangency is the center of the sphere; this form is unique up to congruences. If we reciprocate such a canonical polyhedron about its midsphere, the dual polyhedron will share the same edge-tangency points and so must be canonical, it is the canonical dual, the two together form a canonical dual pair.
When a pair of polyhedra cannot be obtained by reciprocation from each other, they may be called duals of each other as long as the vertices of one correspond to the faces of the other, the edges of one correspond to the edges of the other, in an incidence-preserving way. Such pairs of polyhedra are abstractly dual; the vertices and edges of a convex polyhedron form a graph, embedded on a topological sphere, the surface of the polyhedron. The same graph can be projected to form
Vertex (geometry)
In geometry, a vertex is a point where two or more curves, lines, or edges meet. As a consequence of this definition, the point where two lines meet to form an angle and the corners of polygons and polyhedra are vertices; the vertex of an angle is the point where two rays begin or meet, where two line segments join or meet, where two lines intersect, or any appropriate combination of rays and lines that result in two straight "sides" meeting at one place. A vertex is a corner point of a polygon, polyhedron, or other higher-dimensional polytope, formed by the intersection of edges, faces or facets of the object. In a polygon, a vertex is called "convex" if the internal angle of the polygon, that is, the angle formed by the two edges at the vertex, with the polygon inside the angle, is less than π radians. More a vertex of a polyhedron or polytope is convex if the intersection of the polyhedron or polytope with a sufficiently small sphere centered at the vertex is convex, concave otherwise. Polytope vertices are related to vertices of graphs, in that the 1-skeleton of a polytope is a graph, the vertices of which correspond to the vertices of the polytope, in that a graph can be viewed as a 1-dimensional simplicial complex the vertices of which are the graph's vertices.
However, in graph theory, vertices may have fewer than two incident edges, not allowed for geometric vertices. There is a connection between geometric vertices and the vertices of a curve, its points of extreme curvature: in some sense the vertices of a polygon are points of infinite curvature, if a polygon is approximated by a smooth curve there will be a point of extreme curvature near each polygon vertex. However, a smooth curve approximation to a polygon will have additional vertices, at the points where its curvature is minimal. A vertex of a plane tiling or tessellation is a point. More a tessellation can be viewed as a kind of topological cell complex, as can the faces of a polyhedron or polytope. A polygon vertex xi of a simple polygon P is a principal polygon vertex if the diagonal intersects the boundary of P only at x and x. There are two types of principal vertices: mouths. A principal vertex xi of a simple polygon P is called an ear if the diagonal that bridges xi lies in P. According to the two ears theorem, every simple polygon has at least two ears.
A principal vertex xi of a simple polygon P is called a mouth if the diagonal lies outside the boundary of P. Any convex polyhedron's surface has Euler characteristic V − E + F = 2, where V is the number of vertices, E is the number of edges, F is the number of faces; this equation is known as Euler's polyhedron formula. Thus the number of vertices is 2 more than the excess of the number of edges over the number of faces. For example, a cube has 12 edges and 6 faces, hence 8 vertices. In computer graphics, objects are represented as triangulated polyhedra in which the object vertices are associated not only with three spatial coordinates but with other graphical information necessary to render the object such as colors, reflectance properties and surface normal. Weisstein, Eric W. "Polygon Vertex". MathWorld. Weisstein, Eric W. "Polyhedron Vertex". MathWorld. Weisstein, Eric W. "Principal Vertex". MathWorld
Triangle
A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices A, B, C is denoted △ A B C. In Euclidean geometry any three points, when non-collinear, determine a unique triangle and a unique plane. In other words, there is only one plane that contains that triangle, every triangle is contained in some plane. If the entire geometry is only the Euclidean plane, there is only one plane and all triangles are contained in it; this article is about triangles in Euclidean geometry, in particular, the Euclidean plane, except where otherwise noted. Triangles can be classified according to the lengths of their sides: An equilateral triangle has all sides the same length. An equilateral triangle is a regular polygon with all angles measuring 60°. An isosceles triangle has two sides of equal length. An isosceles triangle has two angles of the same measure, namely the angles opposite to the two sides of the same length; some mathematicians define an isosceles triangle to have two equal sides, whereas others define an isosceles triangle as one with at least two equal sides.
The latter definition would make all equilateral triangles isosceles triangles. The 45–45–90 right triangle, which appears in the tetrakis square tiling, is isosceles. A scalene triangle has all its sides of different lengths. Equivalently, it has all angles of different measure. Hatch marks called tick marks, are used in diagrams of triangles and other geometric figures to identify sides of equal lengths. A side can be marked with a pattern of short line segments in the form of tally marks. In a triangle, the pattern is no more than 3 ticks. An equilateral triangle has the same pattern on all 3 sides, an isosceles triangle has the same pattern on just 2 sides, a scalene triangle has different patterns on all sides since no sides are equal. Patterns of 1, 2, or 3 concentric arcs inside the angles are used to indicate equal angles. An equilateral triangle has the same pattern on all 3 angles, an isosceles triangle has the same pattern on just 2 angles, a scalene triangle has different patterns on all angles since no angles are equal.
Triangles can be classified according to their internal angles, measured here in degrees. A right triangle has one of its interior angles measuring 90°; the side opposite to the right angle is the longest side of the triangle. The other two sides are called the catheti of the triangle. Right triangles obey the Pythagorean theorem: the sum of the squares of the lengths of the two legs is equal to the square of the length of the hypotenuse: a2 + b2 = c2, where a and b are the lengths of the legs and c is the length of the hypotenuse. Special right triangles are right triangles with additional properties that make calculations involving them easier. One of the two most famous is the 3–4–5 right triangle, where 32 + 42 = 52. In this situation, 3, 4, 5 are a Pythagorean triple; the other one is an isosceles triangle. Triangles that do not have an angle measuring 90° are called oblique triangles. A triangle with all interior angles measuring less than 90° is an acute triangle or acute-angled triangle.
If c is the length of the longest side a2 + b2 > c2, where a and b are the lengths of the other sides. A triangle with one interior angle measuring more than 90° is an obtuse triangle or obtuse-angled triangle. If c is the length of the longest side a2 + b2 < c2, where a and b are the lengths of the other sides. A triangle with an interior angle of 180° is degenerate. A right degenerate triangle has collinear vertices. A triangle that has two angles with the same measure has two sides with the same length, therefore it is an isosceles triangle, it follows that in a triangle where all angles have the same measure, all three sides have the same length, such a triangle is therefore equilateral. Triangles are assumed to be two-dimensional plane figures. In rigorous treatments, a triangle is therefore called a 2-simplex. Elementary facts about triangles were presented by Euclid in books 1–4 of his Elements, around 300 BC; the sum of the measures of the interior angles of a triangle in Euclidean space is always 180 degrees.
This fact is equivalent to Euclid's parallel postulate. This allows determination of the measure of the third angle of any triangle given the measure of two angles. An exterior angle of a triangle is an angle, a linear pair to an interior angle; the measure of an exterior angle of a triangle is equal to the sum of the measures of the two interior angles that are not adjacent to it. The sum of the measures of the three exterior angles of any triangle is 360 degrees. Two triangles are said to be similar if every angle of one triangle has the same measure as the corresponding angle in the other triangle; the corresponding sides of similar triangles have lengths that are in the same proportion, this property is sufficient to establish similarity. Some basic theorems about similar triangles are: If and only if one pair of internal angles of two triangles have the sam
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
Truncated hexagonal tiling
In geometry, the truncated hexagonal tiling is a semiregular tiling of the Euclidean plane. There are one triangle on each vertex; as the name implies this tiling is constructed by a truncation operation applies to a hexagonal tiling, leaving dodecagons in place of the original hexagons, new triangles at the original vertex locations. It is given an extended Schläfli symbol of t. Conway calls it a truncated hextille, constructed as a truncation operation applied to a hexagonal tiling. There are 3 regular and 8 semiregular tilings in the plane. There is only one uniform coloring of a truncated hexagonal tiling; the dodecagonal faces can be distorted into different geometries, like: Like the uniform polyhedra there are eight uniform tilings that can be based from the regular hexagonal tiling. Drawing the tiles colored as red on the original faces, yellow at the original vertices, blue along the original edges, there are 8 forms, 7 which are topologically distinct; this tiling is topologically related as a part of sequence of uniform truncated polyhedra with vertex configurations, Coxeter group symmetry.
Two 2-uniform tilings are related by dissected the dodecagons into a central hexagonal and 6 surrounding triangles and squares. The truncated hexagonal tiling can be used as a circle packing, placing equal diameter circles at the center of every point; every circle is in contact with 3 other circles in the packing. This is the lowest density packing; the triakis triangular tiling is a tiling of the Euclidean plane. It is an equilateral triangular tiling with each triangle divided into three obtuse triangles from the center point, it is labeled by face configuration V3.12.12 because each isosceles triangle face has two types of vertices: one with 3 triangles, two with 12 triangles. Conway calls it a kisdeltille, constructed as a kis operation applied to a triangular tiling. In Japan the pattern is called asanoha for hemp leaf, although the name applies to other triakis shapes like the triakis icosahedron and triakis octahedron, it is the dual tessellation of the truncated hexagonal tiling which has one triangle and two dodecagons at each vertex.
It is one including the regular duals. Tilings of regular polygons List of uniform tilings John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 Grünbaum, Branko. C.. Tilings and Patterns. New York: W. H. Freeman. ISBN 0-7167-1193-1. CS1 maint: Multiple names: authors list Williams, Robert; the Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. p. 39. ISBN 0-486-23729-X. Keith Critchlow, Order in Space: A design source book, 1970, p. 69-61, Pattern E, Dual p. 77-76, pattern 1 Dale Seymour and Jill Britton, Introduction to Tessellations, 1989, ISBN 978-0866514613, pp. 50–56, dual p. 117 Weisstein, Eric W. "Semiregular tessellation". MathWorld. Klitzing, Richard. "2D Euclidean tilings o3x6x - toxat - O7"
Geometry
Geometry is a branch of mathematics concerned with questions of shape, relative position of figures, the properties of space. A mathematician who works in the field of geometry is called a geometer. Geometry arose independently in a number of early cultures as a practical way for dealing with lengths and volumes. Geometry began to see elements of formal mathematical science emerging in the West as early as the 6th century BC. By the 3rd century BC, geometry was put into an axiomatic form by Euclid, whose treatment, Euclid's Elements, set a standard for many centuries to follow. Geometry arose independently in India, with texts providing rules for geometric constructions appearing as early as the 3rd century BC. Islamic scientists expanded on them during the Middle Ages. By the early 17th century, geometry had been put on a solid analytic footing by mathematicians such as René Descartes and Pierre de Fermat. Since and into modern times, geometry has expanded into non-Euclidean geometry and manifolds, describing spaces that lie beyond the normal range of human experience.
While geometry has evolved throughout the years, there are some general concepts that are more or less fundamental to geometry. These include the concepts of points, planes, surfaces and curves, as well as the more advanced notions of manifolds and topology or metric. Geometry has applications to many fields, including art, physics, as well as to other branches of mathematics. Contemporary geometry has many subfields: Euclidean geometry is geometry in its classical sense; the mandatory educational curriculum of the majority of nations includes the study of points, planes, triangles, similarity, solid figures and analytic geometry. Euclidean geometry has applications in computer science and various branches of modern mathematics. Differential geometry uses techniques of linear algebra to study problems in geometry, it has applications in physics, including in general relativity. Topology is the field concerned with the properties of geometric objects that are unchanged by continuous mappings. In practice, this means dealing with large-scale properties of spaces, such as connectedness and compactness.
Convex geometry investigates convex shapes in the Euclidean space and its more abstract analogues using techniques of real analysis. It has close connections to convex analysis and functional analysis and important applications in number theory. Algebraic geometry studies geometry through the use of multivariate polynomials and other algebraic techniques, it has applications including cryptography and string theory. Discrete geometry is concerned with questions of relative position of simple geometric objects, such as points and circles, it shares many principles with combinatorics. Computational geometry deals with algorithms and their implementations for manipulating geometrical objects. Although being a young area of geometry, it has many applications in computer vision, image processing, computer-aided design, medical imaging, etc; the earliest recorded beginnings of geometry can be traced to ancient Mesopotamia and Egypt in the 2nd millennium BC. Early geometry was a collection of empirically discovered principles concerning lengths, angles and volumes, which were developed to meet some practical need in surveying, construction and various crafts.
The earliest known texts on geometry are the Egyptian Rhind Papyrus and Moscow Papyrus, the Babylonian clay tablets such as Plimpton 322. For example, the Moscow Papyrus gives a formula for calculating the volume of a truncated pyramid, or frustum. Clay tablets demonstrate that Babylonian astronomers implemented trapezoid procedures for computing Jupiter's position and motion within time-velocity space; these geometric procedures anticipated the Oxford Calculators, including the mean speed theorem, by 14 centuries. South of Egypt the ancient Nubians established a system of geometry including early versions of sun clocks. In the 7th century BC, the Greek mathematician Thales of Miletus used geometry to solve problems such as calculating the height of pyramids and the distance of ships from the shore, he is credited with the first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales' Theorem. Pythagoras established the Pythagorean School, credited with the first proof of the Pythagorean theorem, though the statement of the theorem has a long history.
Eudoxus developed the method of exhaustion, which allowed the calculation of areas and volumes of curvilinear figures, as well as a theory of ratios that avoided the problem of incommensurable magnitudes, which enabled subsequent geometers to make significant advances. Around 300 BC, geometry was revolutionized by Euclid, whose Elements considered the most successful and influential textbook of all time, introduced mathematical rigor through the axiomatic method and is the earliest example of the format still used in mathematics today, that of definition, axiom and proof. Although most of the contents of the Elements were known, Euclid arranged them into a single, coherent logical framework; the Elements was known to all educated people in the West until the middle of the 20th century and its contents are still taught in geometry classes today. Archimedes of Syracuse used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, gave remarkably accurate approximations of Pi.
He studied the sp