1.
Space
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Space is the boundless three-dimensional extent in which objects and events have relative position and direction. Physical space is conceived in three linear dimensions, although modern physicists usually consider it, with time, to be part of a boundless four-dimensional continuum known as spacetime. The concept of space is considered to be of importance to an understanding of the physical universe. However, disagreement continues between philosophers over whether it is itself an entity, a relationship between entities, or part of a conceptual framework. Many of these classical philosophical questions were discussed in the Renaissance and then reformulated in the 17th century, in Isaac Newtons view, space was absolute—in the sense that it existed permanently and independently of whether there was any matter in the space. Other natural philosophers, notably Gottfried Leibniz, thought instead that space was in fact a collection of relations between objects, given by their distance and direction from one another. In the 18th century, the philosopher and theologian George Berkeley attempted to refute the visibility of spatial depth in his Essay Towards a New Theory of Vision. Kant referred to the experience of space in his Critique of Pure Reason as being a pure a priori form of intuition. In the 19th and 20th centuries mathematicians began to examine geometries that are non-Euclidean, in space is conceived as curved. According to Albert Einsteins theory of relativity, space around gravitational fields deviates from Euclidean space. Experimental tests of general relativity have confirmed that non-Euclidean geometries provide a model for the shape of space. In the seventeenth century, the philosophy of space and time emerged as an issue in epistemology. At its heart, Gottfried Leibniz, the German philosopher-mathematician, and Isaac Newton, unoccupied regions are those that could have objects in them, and thus spatial relations with other places. For Leibniz, then, space was an abstraction from the relations between individual entities or their possible locations and therefore could not be continuous but must be discrete. Space could be thought of in a way to the relations between family members. Although people in the family are related to one another, the relations do not exist independently of the people, but since there would be no observational way of telling these universes apart then, according to the identity of indiscernibles, there would be no real difference between them. According to the principle of sufficient reason, any theory of space that implied that there could be two possible universes must therefore be wrong. Newton took space to be more than relations between objects and based his position on observation and experimentation
2.
Geometry
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Geometry is a branch of mathematics concerned with questions of shape, size, relative position of figures, and 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, areas, and volumes. Geometry began to see elements of mathematical science emerging in the West as early as the 6th century BC. By the 3rd century BC, geometry was put into a form by Euclid, whose treatment, Euclids Elements. Geometry arose independently in India, with texts providing rules for geometric constructions appearing as early as the 3rd century BC, islamic scientists preserved Greek ideas and expanded on them during the Middle Ages. By the early 17th century, geometry had been put on a solid footing by mathematicians such as René Descartes. Since then, and into modern times, geometry has expanded into non-Euclidean geometry and manifolds, while geometry has evolved significantly throughout the years, there are some general concepts that are more or less fundamental to geometry. These include the concepts of points, lines, planes, surfaces, angles, 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, lines, planes, angles, triangles, congruence, similarity, solid figures, circles, Euclidean geometry also has applications in computer science, crystallography, and various branches of modern mathematics. Differential geometry uses techniques of calculus and linear algebra to 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 often means dealing with large-scale properties of spaces, convex geometry investigates convex shapes in the Euclidean space and its more abstract analogues, often using techniques of real analysis. It has close connections to convex analysis, optimization and functional analysis, algebraic geometry studies geometry through the use of multivariate polynomials and other algebraic techniques. It has applications in areas, including cryptography and string theory. Discrete geometry is concerned mainly with questions of relative position of simple objects, such as points. It shares many methods and principles with combinatorics, Geometry has applications to many fields, including art, architecture, physics, as well as to other branches of mathematics. The earliest recorded beginnings of geometry can be traced to ancient Mesopotamia, 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, later clay tablets demonstrate that Babylonian astronomers implemented trapezoid procedures for computing Jupiters position and motion within time-velocity space
3.
Euclidean geometry
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Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry, the Elements. Euclids method consists in assuming a set of intuitively appealing axioms. Although many of Euclids results had been stated by earlier mathematicians, Euclid was the first to show how these propositions could fit into a comprehensive deductive and logical system. The Elements begins with plane geometry, still taught in school as the first axiomatic system. It goes on to the geometry of three dimensions. Much of the Elements states results of what are now called algebra and number theory, for more than two thousand years, the adjective Euclidean was unnecessary because no other sort of geometry had been conceived. Euclids axioms seemed so obvious that any theorem proved from them was deemed true in an absolute, often metaphysical. Today, however, many other self-consistent non-Euclidean geometries are known, Euclidean geometry is an example of synthetic geometry, in that it proceeds logically from axioms to propositions without the use of coordinates. This is in contrast to analytic geometry, which uses coordinates, the Elements is mainly a systematization of earlier knowledge of geometry. Its improvement over earlier treatments was recognized, with the result that there was little interest in preserving the earlier ones. There are 13 total books in the Elements, Books I–IV, Books V and VII–X deal with number theory, with numbers treated geometrically via their representation as line segments with various lengths. Notions such as numbers and rational and irrational numbers are introduced. The infinitude of prime numbers is proved, a typical result is the 1,3 ratio between the volume of a cone and a cylinder with the same height and base. Euclidean geometry is a system, in which all theorems are derived from a small number of axioms. To produce a straight line continuously in a straight line. To describe a circle with any centre and distance and that all right angles are equal to one another. Although Euclids statement of the only explicitly asserts the existence of the constructions. The Elements also include the five common notions, Things that are equal to the same thing are also equal to one another
4.
Fractal
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A fractal is a mathematical set that exhibits a repeating pattern displayed at every scale. It is also known as expanding symmetry or evolving symmetry, if the replication is exactly the same at every scale, it is called a self-similar pattern. An example of this is the Menger Sponge, Fractals can also be nearly the same at different levels. This latter pattern is illustrated in small magnifications of the Mandelbrot set, Fractals also include the idea of a detailed pattern that repeats itself. Fractals are different from other geometric figures because of the way in which they scale, doubling the edge lengths of a polygon multiplies its area by four, which is two raised to the power of two. Likewise, if the radius of a sphere is doubled, its volume scales by eight, but if a fractals one-dimensional lengths are all doubled, the spatial content of the fractal scales by a power that is not necessarily an integer. This power is called the dimension of the fractal. As mathematical equations, fractals are usually nowhere differentiable, the term fractal was first used by mathematician Benoît Mandelbrot in 1975. Mandelbrot based it on the Latin frāctus meaning broken or fractured, there is some disagreement amongst authorities about how the concept of a fractal should be formally defined. Mandelbrot himself summarized it as beautiful, damn hard, increasingly useful, Fractals are not limited to geometric patterns, but can also describe processes in time. Fractal patterns with various degrees of self-similarity have been rendered or studied in images, structures and sounds and found in nature, technology, art, Fractals are of particular relevance in the field of chaos theory, since the graphs of most chaotic processes are fractal. The word fractal often has different connotations for laypeople than for mathematicians, the mathematical concept is difficult to define formally even for mathematicians, but key features can be understood with little mathematical background. If this is done on fractals, however, no new detail appears, nothing changes, self-similarity itself is not necessarily counter-intuitive. The difference for fractals is that the pattern reproduced must be detailed, a regular line, for instance, is conventionally understood to be 1-dimensional, if such a curve is divided into pieces each 1/3 the length of the original, there are always 3 equal pieces. In contrast, consider the Koch snowflake and it is also 1-dimensional for the same reason as the ordinary line, but it has, in addition, a fractal dimension greater than 1 because of how its detail can be measured. This also leads to understanding a third feature, that fractals as mathematical equations are nowhere differentiable, in a concrete sense, this means fractals cannot be measured in traditional ways. The history of fractals traces a path from chiefly theoretical studies to modern applications in computer graphics, according to Pickover, the mathematics behind fractals began to take shape in the 17th century when the mathematician and philosopher Gottfried Leibniz pondered recursive self-similarity. In his writings, Leibniz used the term fractional exponents, also in the last part of that century, Felix Klein and Henri Poincaré introduced a category of fractal that has come to be called self-inverse fractals
5.
Symmetry
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Symmetry in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, symmetry has a precise definition, that an object is invariant to any of various transformations. Although these two meanings of symmetry can sometimes be told apart, they are related, so they are discussed together. The opposite of symmetry is asymmetry, a geometric shape or object is symmetric if it can be divided into two or more identical pieces that are arranged in an organized fashion. This means that an object is symmetric if there is a transformation that moves individual pieces of the object, an object has rotational symmetry if the object can be rotated about a fixed point without changing the overall shape. An object has symmetry if it can be translated without changing its overall shape. An object has symmetry if it can be simultaneously translated and rotated in three-dimensional space along a line known as a screw axis. An object has symmetry if it does not change shape when it is expanded or contracted. Fractals also exhibit a form of symmetry, where small portions of the fractal are similar in shape to large portions. Other symmetries include glide reflection symmetry and rotoreflection symmetry, a dyadic relation R is symmetric if and only if, whenever its true that Rab, its true that Rba. Thus, is the age as is symmetrical, for if Paul is the same age as Mary. Symmetric binary logical connectives are and, or, biconditional, nand, xor, the set of operations that preserve a given property of the object form a group. In general, every kind of structure in mathematics will have its own kind of symmetry, examples include even and odd functions in calculus, the symmetric group in abstract algebra, symmetric matrices in linear algebra, and the Galois group in Galois theory. In statistics, it appears as symmetric probability distributions, and as skewness, symmetry in physics has been generalized to mean invariance—that is, lack of change—under any kind of transformation, for example arbitrary coordinate transformations. This concept has one of the most powerful tools of theoretical physics. See Noethers theorem, and also, Wigners classification, which says that the symmetries of the laws of physics determine the properties of the found in nature. Important symmetries in physics include continuous symmetries and discrete symmetries of spacetime, internal symmetries of particles, in biology, the notion of symmetry is mostly used explicitly to describe body shapes. Bilateral animals, including humans, are more or less symmetric with respect to the plane which divides the body into left
6.
Hyperbolic geometry
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In mathematics, hyperbolic geometry is a non-Euclidean geometry. Hyperbolic plane geometry is also the geometry of saddle surface or pseudospherical surfaces, surfaces with a constant negative Gaussian curvature, a modern use of hyperbolic geometry is in the theory of special relativity, particularly Minkowski spacetime and gyrovector space. In Russia it is commonly called Lobachevskian geometry, named one of its discoverers. This page is mainly 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, see hyperbolic space for more on the three and higher dimensional cases. Hyperbolic geometry is closely related to Euclidean geometry than it seems. When the parallel postulate is removed from Euclidean geometry the resulting geometry is absolute geometry, there are two kinds of absolute geometry, Euclidean and hyperbolic. All theorems of geometry, including the first 28 propositions of book one of Euclids Elements, are valid in Euclidean. Propositions 27 and 28 of Book One of Euclids Elements prove the existence of parallel/non-intersecting lines and this difference also has many consequences, concepts that are equivalent in Euclidean geometry are not equivalent in hyperbolic geometry, new concepts need to be introduced. Further, because of the angle of parallelism hyperbolic geometry has an absolute scale, single lines in hyperbolic geometry have exactly the same properties as single straight lines in Euclidean geometry. For example, two points define a line, and lines can be infinitely extended. Two intersecting lines have the 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, when we add a third line then there are properties of intersecting lines that differ from intersecting lines in Euclidean geometry. For example, given 2 intersecting lines there are many lines that do not intersect either of the given lines. While in some models lines look different they do have these properties, non-intersecting lines in hyperbolic geometry also 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 lines through P that do not intersect R. This implies that there are through P an infinite number of lines that do not intersect R. All other non-intersecting lines have a point of distance and diverge from both sides of that point, and are called ultraparallel, diverging parallel or sometimes non-intersecting. Some geometers simply use parallel lines instead of limiting parallel lines and these limiting parallels make an angle θ with PB, this angle depends only on the Gaussian curvature of the plane and the distance PB and is called the angle of parallelism
7.
History of geometry
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Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of mathematics, the other being the study of numbers. Classic geometry was focused in compass and straightedge constructions, geometry was revolutionized by Euclid, who introduced mathematical rigor and the axiomatic method still in use today. His book, The Elements is widely considered the most influential textbook of all time, the earliest recorded beginnings of geometry can be traced to early peoples, who discovered obtuse triangles in the ancient Indus Valley, and ancient Babylonia from around 3000 BC. Among these were some surprisingly sophisticated principles, and a mathematician might be hard put to derive some of them without the use of calculus. For example, both the Egyptians and the Babylonians were aware of versions of the Pythagorean theorem about 1500 years before Pythagoras and the Indian Sulba Sutras around 800 B. C. Problem 30 of the Ahmes papyrus uses these methods to calculate the area of a circle and this assumes that π is 4×², with an error of slightly over 0.63 percent. Problem 48 involved using a square with side 9 units and this square was cut into a 3x3 grid. The diagonal of the squares were used to make an irregular octagon with an area of 63 units. This gave a value for π of 3.111. The two problems together indicate a range of values for π between 3.11 and 3.16. Problem 14 in the Moscow Mathematical Papyrus gives the only ancient example finding the volume of a frustum of a pyramid, describing the correct formula, the Babylonians may have known the general rules for measuring areas and volumes. They measured the circumference of a circle as three times the diameter and the area as one-twelfth the square of the circumference, which would be correct if π is estimated as 3, the Pythagorean theorem was also known to the Babylonians. Also, there was a recent discovery in which a tablet used π as 3, the Babylonians are also known for the Babylonian mile, which was a measure of distance equal to about seven miles today. This measurement for distances eventually was converted to a used for measuring the travel of the Sun, therefore. There have been recent discoveries showing that ancient Babylonians may have discovered astronomical geometry nearly 1400 years before Europeans did, the Indian Vedic period had a tradition of geometry, mostly expressed in the construction of elaborate altars. Early Indian texts on this include the Satapatha Brahmana and the Śulba Sūtras. According to, the Śulba Sūtras contain the earliest extant verbal expression of the Pythagorean Theorem in the world, the diagonal rope of an oblong produces both which the flank and the horizontal <ropes> produce separately
8.
Sphere
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A sphere is a perfectly round geometrical object in three-dimensional space that is the surface of a completely round ball. This distance r is the radius of the ball, and the point is the center of the mathematical ball. The longest straight line through the ball, connecting two points of the sphere, passes through the center and its length is twice the radius. While outside mathematics the terms sphere and ball are used interchangeably. The ball and the share the same radius, diameter. The surface area of a sphere is, A =4 π r 2, at any given radius r, the incremental volume equals the product of the surface area at radius r and the thickness of a shell, δ V ≈ A ⋅ δ r. The total volume is the summation of all volumes, V ≈ ∑ A ⋅ δ r. In the limit as δr approaches zero this equation becomes, V = ∫0 r A d r ′, substitute V,43 π r 3 = ∫0 r A d r ′. Differentiating both sides of equation with respect to r yields A as a function of r,4 π r 2 = A. Which is generally abbreviated as, A =4 π r 2, alternatively, the area element on the sphere is given in spherical coordinates by dA = r2 sin θ dθ dφ. In Cartesian coordinates, the element is d S = r r 2 − ∑ i ≠ k x i 2 ∏ i ≠ k d x i, ∀ k. For more generality, see area element, the total area can thus be obtained by integration, A = ∫02 π ∫0 π r 2 sin θ d θ d φ =4 π r 2. In three dimensions, the volume inside a sphere is derived to be V =43 π r 3 where r is the radius of the sphere, archimedes first derived this formula, which shows that the volume inside a sphere is 2/3 that of a circumscribed cylinder. In modern mathematics, this formula can be derived using integral calculus, at any given x, the incremental volume equals the product of the cross-sectional area of the disk at x and its thickness, δ V ≈ π y 2 ⋅ δ x. The total volume is the summation of all volumes, V ≈ ∑ π y 2 ⋅ δ x. In the limit as δx approaches zero this equation becomes, V = ∫ − r r π y 2 d x. At any given x, a right-angled triangle connects x, y and r to the origin, hence, applying the Pythagorean theorem yields, thus, substituting y with a function of x gives, V = ∫ − r r π d x. Which can now be evaluated as follows, V = π − r r = π − π =43 π r 3
9.
Rectangle
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In Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as a quadrilateral, since equiangular means that all of its angles are equal. It can also be defined as a parallelogram containing a right angle, a rectangle with four sides of equal length is a square. The term oblong is occasionally used to refer to a non-square rectangle, a rectangle with vertices ABCD would be denoted as ABCD. The word rectangle comes from the Latin rectangulus, which is a combination of rectus and angulus, a crossed rectangle is a crossed quadrilateral which consists of two opposite sides of a rectangle along with the two diagonals. It is a case of an antiparallelogram, and its angles are not right angles. Other geometries, such as spherical, elliptic, and hyperbolic, have so-called rectangles with sides equal in length. Rectangles are involved in many tiling problems, such as tiling the plane by rectangles or tiling a rectangle by polygons, a convex quadrilateral with successive sides a, b, c, d whose area is 12. A rectangle is a case of a parallelogram in which each pair of adjacent sides is perpendicular. A parallelogram is a case of a trapezium in which both pairs of opposite sides are parallel and equal in length. A trapezium is a quadrilateral which has at least one pair of parallel opposite sides. A convex quadrilateral is Simple, The boundary does not cross itself, star-shaped, The whole interior is visible from a single point, without crossing any edge. De Villiers defines a more generally as any quadrilateral with axes of symmetry through each pair of opposite sides. This definition includes both right-angled rectangles and crossed rectangles, quadrilaterals with two axes of symmetry, each through a pair of opposite sides, belong to the larger class of quadrilaterals with at least one axis of symmetry through a pair of opposite sides. These quadrilaterals comprise isosceles trapezia and crossed isosceles trapezia, a rectangle is cyclic, all corners lie on a single circle. It is equiangular, all its corner angles are equal and it is isogonal or vertex-transitive, all corners lie within the same symmetry orbit. It has two lines of symmetry and rotational symmetry of order 2. The dual polygon of a rectangle is a rhombus, as shown in the table below, the figure formed by joining, in order, the midpoints of the sides of a rectangle is a rhombus and vice versa
10.
Rhombus
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In Euclidean geometry, a rhombus is a simple quadrilateral whose four sides all have the same length. Another name is equilateral quadrilateral, since equilateral means that all of its sides are equal in length, every rhombus is a parallelogram and a kite. A rhombus with right angles is a square, the word rhombus comes from Greek ῥόμβος, meaning something that spins, which derives from the verb ῥέμβω, meaning to turn round and round. The word was used both by Euclid and Archimedes, who used the term solid rhombus for two right circular cones sharing a common base, the surface we refer to as rhombus today is a cross section of this solid rhombus through the apex of each of the two cones. This is a case of the superellipse, with exponent 1. Every rhombus has two diagonals connecting pairs of vertices, and two pairs of parallel sides. Using congruent triangles, one can prove that the rhombus is symmetric across each of these diagonals and it follows that any rhombus has the following properties, Opposite angles of a rhombus have equal measure. The two diagonals of a rhombus are perpendicular, that is, a rhombus is an orthodiagonal quadrilateral, the first property implies that every rhombus is a parallelogram. Thus denoting the common side as a and the diagonals as p and q, not every parallelogram is a rhombus, though any parallelogram with perpendicular diagonals is a rhombus. In general, any quadrilateral with perpendicular diagonals, one of which is a line of symmetry, is a kite, every rhombus is a kite, and any quadrilateral that is both a kite and parallelogram is a rhombus. A rhombus is a tangential quadrilateral and that is, it has an inscribed circle that is tangent to all four sides. As for all parallelograms, the area K of a rhombus is the product of its base, the base is simply any side length a, K = a ⋅ h. The inradius, denoted by r, can be expressed in terms of the p and q as. The dual polygon of a rhombus is a rectangle, A rhombus has all sides equal, a rhombus has opposite angles equal, while a rectangle has opposite sides equal. A rhombus has a circle, while a rectangle has a circumcircle. A rhombus has an axis of symmetry through each pair of opposite vertex angles, the diagonals of a rhombus intersect at equal angles, while the diagonals of a rectangle are equal in length. The figure formed by joining the midpoints of the sides of a rhombus is a rectangle, a rhombohedron is a three-dimensional figure like a cube, except that its six faces are rhombi instead of squares. The rhombic dodecahedron is a polyhedron with 12 congruent rhombi as its faces
11.
Compass-and-straightedge construction
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The idealized ruler, known as a straightedge, is assumed to be infinite in length, and has no markings on it and only one edge. The compass is assumed to collapse when lifted from the page, more formally, the only permissible constructions are those granted by Euclids first three postulates. It turns out to be the case that every point constructible using straightedge, the ancient Greek mathematicians first conceived compass-and-straightedge constructions, and a number of ancient problems in plane geometry impose this restriction. The ancient Greeks developed many constructions, but in cases were unable to do so. Gauss showed that some polygons are constructible but that most are not, some of the most famous straightedge-and-compass problems were proven impossible by Pierre Wantzel in 1837, using the mathematical theory of fields. In spite of existing proofs of impossibility, some persist in trying to solve these problems, in terms of algebra, a length is constructible if and only if it represents a constructible number, and an angle is constructible if and only if its cosine is a constructible number. A number is constructible if and only if it can be using the four basic arithmetic operations. Circles can only be starting from two given points, the centre and a point on the circle. The compass may or may not collapse when its not drawing a circle, the straightedge is infinitely long, but it has no markings on it and has only one straight edge, unlike ordinary rulers. It can only be used to draw a segment between two points or to extend an existing segment. The modern compass generally does not collapse and several modern constructions use this feature and it would appear that the modern compass is a more powerful instrument than the ancient collapsing compass. However, by Proposition 2 of Book 1 of Euclids Elements, although the proposition is correct, its proofs have a long and checkered history. Eyeballing it and getting close does not count as a solution and that is, it must have a finite number of steps, and not be the limit of ever closer approximations. One of the purposes of Greek mathematics was to find exact constructions for various lengths, for example. The Greeks could not find constructions for these three problems, among others, Squaring the circle, Drawing a square the same area as a given circle, doubling the cube, Drawing a cube with twice the volume of a given cube. Trisecting the angle, Dividing a given angle into three smaller angles all of the same size, for 2000 years people tried to find constructions within the limits set above, and failed. All three have now been proven under mathematical rules to be generally impossible, the ancient Greek mathematicians first attempted compass-and-straightedge constructions, and they discovered how to construct sums, differences, products, ratios, and square roots of given lengths. They could also construct half of an angle, a square whose area is twice that of another square, a square having the same area as a given polygon
12.
Polygon
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In elementary geometry, a polygon /ˈpɒlɪɡɒn/ is a plane figure that is bounded by a finite chain of straight line segments closing in a loop to form a closed polygonal chain or circuit. These segments are called its edges or sides, and the points where two edges meet are the vertices or corners. The interior of the polygon is called its body. An n-gon is a polygon with n sides, for example, a polygon is a 2-dimensional example of the more general polytope in any number of dimensions. The basic geometrical notion of a polygon has been adapted in various ways to suit particular purposes, mathematicians are often concerned only with the bounding closed polygonal chain and with simple polygons which do not self-intersect, and they often define a polygon accordingly. A polygonal boundary may be allowed to intersect itself, creating star polygons and these and other generalizations of polygons are described below. The word polygon derives from the Greek adjective πολύς much, many and it has been suggested that γόνυ knee may be the origin of “gon”. Polygons are primarily classified by the number of sides, Polygons may be characterized by their convexity or type of non-convexity, Convex, any line drawn through the polygon meets its boundary exactly twice. As a consequence, all its interior angles are less than 180°, equivalently, any line segment with endpoints on the boundary passes through only interior points between its endpoints. Non-convex, a line may be found which meets its boundary more than twice, equivalently, there exists a line segment between two boundary points that passes outside the polygon. Simple, the boundary of the polygon does not cross itself, there is at least one interior angle greater than 180°. Star-shaped, the interior is visible from at least one point. The polygon must be simple, and may be convex or concave, self-intersecting, the boundary of the polygon crosses itself. Branko Grünbaum calls these coptic, though this term does not seem to be widely used, star polygon, a polygon which self-intersects in a regular way. A polygon cannot be both a star and star-shaped, equiangular, all corner angles are equal. Cyclic, all lie on a single circle, called the circumcircle. Isogonal or vertex-transitive, all lie within the same symmetry orbit. The polygon is cyclic and equiangular