In three-dimensional space, a Platonic solid is a regular, convex polyhedron. It is constructed by congruent regular polygonal faces with the same number of faces meeting at each vertex. Five solids meet these criteria: Geometers have studied the Platonic solids for thousands of years, they are named for the ancient Greek philosopher Plato who hypothesized in his dialogue, the Timaeus, that the classical elements were made of these regular solids. The Platonic solids have been known since antiquity, it has been suggested that certain carved stone balls created by the late Neolithic people of Scotland represent these shapes. The ancient Greeks studied the Platonic solids extensively; some sources credit Pythagoras with their discovery. Other evidence suggests that he may have only been familiar with the tetrahedron and dodecahedron and that the discovery of the octahedron and icosahedron belong to Theaetetus, a contemporary of Plato. In any case, Theaetetus gave a mathematical description of all five and may have been responsible for the first known proof that no other convex regular polyhedra exist.
The Platonic solids are prominent in the philosophy of Plato, their namesake. Plato wrote about them in the dialogue Timaeus c.360 B. C. in which he associated each of the four classical elements with a regular solid. Earth was associated with the cube, air with the octahedron, water with the icosahedron, fire with the tetrahedron. There was intuitive justification for these associations: the heat of fire feels sharp and stabbing. Air is made of the octahedron. Water, the icosahedron, flows out of one's hand when picked up, as if it is made of tiny little balls. By contrast, a nonspherical solid, the hexahedron represents "earth"; these clumsy little solids cause dirt to crumble and break when picked up in stark difference to the smooth flow of water. Moreover, the cube's being the only regular solid that tessellates Euclidean space was believed to cause the solidity of the Earth. Of the fifth Platonic solid, the dodecahedron, Plato obscurely remarks, "...the god used for arranging the constellations on the whole heaven".
Aristotle added a fifth element, aithēr and postulated that the heavens were made of this element, but he had no interest in matching it with Plato's fifth solid. Euclid mathematically described the Platonic solids in the Elements, the last book of, devoted to their properties. Propositions 13–17 in Book XIII describe the construction of the tetrahedron, cube and dodecahedron in that order. For each solid Euclid finds the ratio of the diameter of the circumscribed sphere to the edge length. In Proposition 18 he argues. Andreas Speiser has advocated the view that the construction of the 5 regular solids is the chief goal of the deductive system canonized in the Elements. Much of the information in Book XIII is derived from the work of Theaetetus. In the 16th century, the German astronomer Johannes Kepler attempted to relate the five extraterrestrial planets known at that time to the five Platonic solids. In Mysterium Cosmographicum, published in 1596, Kepler proposed a model of the Solar System in which the five solids were set inside one another and separated by a series of inscribed and circumscribed spheres.
Kepler proposed that the distance relationships between the six planets known at that time could be understood in terms of the five Platonic solids enclosed within a sphere that represented the orbit of Saturn. The six spheres each corresponded to one of the planets; the solids were ordered with the innermost being the octahedron, followed by the icosahedron, dodecahedron and the cube, thereby dictating the structure of the solar system and the distance relationships between the planets by the Platonic solids. In the end, Kepler's original idea had to be abandoned, but out of his research came his three laws of orbital dynamics, the first of, that the orbits of planets are ellipses rather than circles, changing the course of physics and astronomy, he discovered the Kepler solids. In the 20th century, attempts to link Platonic solids to the physical world were expanded to the electron shell model in chemistry by Robert Moon in a theory known as the "Moon model". For Platonic solids centered at the origin, simple Cartesian coordinates of the vertices are given below.
The Greek letter φ is used to represent the golden ratio 1 + √5/2 ≈ 1.6180. The coordinates for the tetrahedron and dodecahedron are given in two orientation sets, each containing half of the sign and position permutation of coordinates; these coordinates reveal certain relationships between the Platonic solids: the vertices of the tetrahedron represent half of those of the cube, as or, one of two sets of 4 vertices in dual positions, as h or. Both tetrahedral positions make the compound stellated octahedron; the coordinates of the icosahedron are related to two alternated sets of coordinates of a nonuniform truncated octahedron, t or called a snub octahedron, as s or, seen in the compound of two icosahedra. Eight of the vertices of the dodecahedron are shared with the cube. Completing all orientat
A prime number is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers. A natural number greater than 1, not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, 1 × 5 or 5 × 1, involve 5 itself. However, 6 is composite because it is the product of two numbers that are both smaller than 6. Primes are central in number theory because of the fundamental theorem of arithmetic: every natural number greater than 1 is either a prime itself or can be factorized as a product of primes, unique up to their order; the property of being prime is called primality. A simple but slow method of checking the primality of a given number n, called trial division, tests whether n is a multiple of any integer between 2 and n. Faster algorithms include the Miller–Rabin primality test, fast but has a small chance of error, the AKS primality test, which always produces the correct answer in polynomial time but is too slow to be practical.
Fast methods are available for numbers of special forms, such as Mersenne numbers. As of December 2018 the largest known prime number has 24,862,048 decimal digits. There are infinitely many primes, as demonstrated by Euclid around 300 BC. No known simple formula separates prime numbers from composite numbers. However, the distribution of primes within the natural numbers in the large can be statistically modelled; the first result in that direction is the prime number theorem, proven at the end of the 19th century, which says that the probability of a randomly chosen number being prime is inversely proportional to its number of digits, that is, to its logarithm. Several historical questions regarding prime numbers are still unsolved; these include Goldbach's conjecture, that every integer greater than 2 can be expressed as the sum of two primes, the twin prime conjecture, that there are infinitely many pairs of primes having just one number between them. Such questions spurred the development of various branches of number theory, focusing on analytic or algebraic aspects of numbers.
Primes are used in several routines in information technology, such as public-key cryptography, which relies on the difficulty of factoring large numbers into their prime factors. In abstract algebra, objects that behave in a generalized way like prime numbers include prime elements and prime ideals. A natural number is called a prime number if it is greater than 1 and cannot be written as a product of two natural numbers that are both smaller than it; the numbers greater than 1 that are not prime are called composite numbers. In other words, n is prime if n items cannot be divided up into smaller equal-size groups of more than one item, or if it is not possible to arrange n dots into a rectangular grid, more than one dot wide and more than one dot high. For example, among the numbers 1 through 6, the numbers 2, 3, 5 are the prime numbers, as there are no other numbers that divide them evenly. 1 is not prime, as it is excluded in the definition. 4 = 2 × 2 and 6 = 2 × 3 are both composite. The divisors of a natural number n are the numbers.
Every natural number has both itself as a divisor. If it has any other divisor, it cannot be prime; this idea leads to a different but equivalent definition of the primes: they are the numbers with two positive divisors, 1 and the number itself. Yet another way to express the same thing is that a number n is prime if it is greater than one and if none of the numbers 2, 3, …, n − 1 divides n evenly; the first 25 prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97. No number n greater than 2 is prime because any such number can be expressed as the product 2 × n / 2. Therefore, every prime number other than 2 is an odd number, is called an odd prime; when written in the usual decimal system, all prime numbers larger than 5 end in 1, 3, 7, or 9. The numbers that end with other digits are all composite: decimal numbers that end in 0, 2, 4, 6, or 8 are and decimal numbers that end in 0 or 5 are divisible by 5; the set of all primes is sometimes denoted by P or by P.
The Rhind Mathematical Papyrus, from around 1550 BC, has Egyptian fraction expansions of different forms for prime and composite numbers. However, the earliest surviving records of the explicit study of prime numbers come from Ancient Greek mathematics. Euclid's Elements proves the infinitude of primes and the fundamental theorem of arithmetic, shows how to construct a perfect number from a Mersenne prime. Another Greek invention, the Sieve of Eratosthenes, is still used to construct lists of primes. Around 1000 AD, the Islamic mathematician Alhazen found Wilson's theorem, characterizing the prime numbers as the numbers n that evenly divide
In geometry, a Kepler–Poinsot polyhedron is any of four regular star polyhedra. They may be obtained by stellating the regular convex dodecahedron and icosahedron, differ from these in having regular pentagrammic faces or vertex figures, they can all be seen as three-dimensional analogues of the pentagram in another. These figures have pentagrams as faces or vertex figures; the small and great stellated dodecahedron have nonconvex regular pentagram faces. The great dodecahedron and great icosahedron have convex polygonal faces, but pentagrammic vertex figures. In all cases, two faces can intersect along a line, not an edge of either face, so that part of each face passes through the interior of the figure; such lines of intersection are not part of the polyhedral structure and are sometimes called false edges. Where three such lines intersect at a point, not a corner of any face, these points are false vertices; the images below show spheres at the true vertices, blue rods along the true edges.
For example, the small stellated dodecahedron has 12 pentagram faces with the central pentagonal part hidden inside the solid. The visible parts of each face comprise five isosceles triangles which touch at five points around the pentagon. We could treat these triangles as 60 separate faces to obtain a new, irregular polyhedron which looks outwardly identical; each edge would now be divided into three shorter edges, the 20 false vertices would become true ones, so that we have a total of 32 vertices. The hidden inner pentagons are no longer part of the polyhedral surface, can disappear. Now Euler's formula holds: 60 − 90 + 32 = 2; however this polyhedron is no longer the one described by the Schläfli symbol, so can not be a Kepler–Poinsot solid though it still looks like one from outside. A Kepler–Poinsot polyhedron covers its circumscribed sphere more than once, with the centers of faces acting as winding points in the figures which have pentagrammic faces, the vertices in the others; because of this, they are not topologically equivalent to the sphere as Platonic solids are, in particular the Euler relation χ = V − E + F = 2 does not always hold.
Schläfli held that all polyhedra must have χ = 2, he rejected the small stellated dodecahedron and great dodecahedron as proper polyhedra. This view was never held. A modified form of Euler's formula, using density of the vertex figures and faces was given by Arthur Cayley, holds both for convex polyhedra, the Kepler–Poinsot polyhedra: d v V − E + d f F = 2 D; the Kepler–Poinsot polyhedra exist in dual pairs. Duals have the same Petrie polygon, or more Petrie polygons with the same two dimensional projection; the following images show the two dual compounds with the same edge radius. They show that the Petrie polygons are skew. Two relationships described in the article below are easily seen in the images: That the violet edges are the same, that the green faces lie in the same planes. John Conway defines the Kepler–Poinsot polyhedra as greatenings and stellations of the convex solids. In his naming convention the small stellated dodecahedron is just the stellated dodecahedron. Stellation changes pentagonal faces into pentagrams.
Greatening maintains the type of faces and resizing them into parallel planes. The great icosahedron is one of the stellations of the icosahedron; the three others are all the stellations of the dodecahedron. The great stellated dodecahedron is a faceting of the dodecahedron; the three others are facetings of the icosahedron. If the intersections are treated as new edges and vertices, the figures obtained will not be regular, but they can still be considered stellations; the great stellated dodecahedron shares its vertices with the dodecahedron. The other three Kepler–Poinsot polyhedra share theirs with the icosahedron; the skeletons of the solids sharing vertices are topologically equivalent. The small and great stellated dodecahedron can be seen as a regular and a great dodecahedron with their edges and faces extended until they intersect; the pentagon faces of these cores are the invisible parts of the star polyhedra's pentagram faces. For the small stellated dodecahedron the hull is φ times bigger than the core, for the great it is φ + 1 = φ 2 times bigger.
Traditionally the two star polyhedra have been defined as augmentations, i.e. as dodecahedron and icosahedron with pyramids added to their faces. Kepler calls the small stellation an augmented dodecahedron. In his view the great stellation is related to the icosahedron as the small one is to the dodecahedron; these naïve definitions are still used. E.g. MathWorld states that the two star polyhedra can be constructed by adding pyramids to the faces of the Platonic solids; this is just a help to visualize the shape of these solids, not a claim that the edg
In mathematical analysis, a measure on a set is a systematic way to assign a number to each suitable subset of that set, intuitively interpreted as its size. In this sense, a measure is a generalization of the concepts of length and volume. A important example is the Lebesgue measure on a Euclidean space, which assigns the conventional length and volume of Euclidean geometry to suitable subsets of the n-dimensional Euclidean space Rn. For instance, the Lebesgue measure of the interval in the real numbers is its length in the everyday sense of the word 1. Technically, a measure is a function that assigns a non-negative real number or +∞ to subsets of a set X, it must further be countably additive: the measure of a'large' subset that can be decomposed into a finite number of'smaller' disjoint subsets is equal to the sum of the measures of the "smaller" subsets. In general, if one wants to associate a consistent size to each subset of a given set while satisfying the other axioms of a measure, one only finds trivial examples like the counting measure.
This problem was resolved by defining measure only on a sub-collection of all subsets. This means that countable unions, countable intersections and complements of measurable subsets are measurable. Non-measurable sets in a Euclidean space, on which the Lebesgue measure cannot be defined are complicated in the sense of being badly mixed up with their complement. Indeed, their existence is a non-trivial consequence of the axiom of choice. Measure theory was developed in successive stages during the late 19th and early 20th centuries by Émile Borel, Henri Lebesgue, Johann Radon, Maurice Fréchet, among others; the main applications of measures are in the foundations of the Lebesgue integral, in Andrey Kolmogorov's axiomatisation of probability theory and in ergodic theory. In integration theory, specifying a measure allows one to define integrals on spaces more general than subsets of Euclidean space. Probability theory considers measures that assign to the whole set the size 1, considers measurable subsets to be events whose probability is given by the measure.
Ergodic theory considers measures that are invariant under, or arise from, a dynamical system. Let X be a set and Σ a σ-algebra over X. A function μ from Σ to the extended real number line is called a measure if it satisfies the following properties: Non-negativity: For all E in Σ: μ ≥ 0. Null empty set: μ = 0. Countable additivity: For all countable collections i = 1 ∞ of pairwise disjoint sets in Σ: μ = ∑ k = 1 ∞ μ One may require that at least one set E has finite measure; the empty set automatically has measure zero because of countable additivity, because μ = μ = μ + μ + μ + …, which implies that μ = 0. If only the second and third conditions of the definition of measure above are met, μ takes on at most one of the values ±∞ μ is called a signed measure; the pair is called a measurable space, the members of Σ are called measurable sets. If and are two measurable spaces a function f: X → Y is called measurable if for every Y-measurable set B ∈ Σ Y, the inverse image is X-measurable – i.e.: f ∈ Σ X.
In this setup, the composition of measurable functions is measurable, making the measurable spaces and measurable functions a category, with the measurable spaces as objects and the set of measurable functions as arrows. See Measurable function#Term usage variations about another setup. A triple is called a measure space. A probability measure is a measure with total measure one – i.e. Μ = 1. A probability space is a measure space with a probability measure. For measure spaces that are topological spaces various compatibility conditions can be
Number theory is a branch of pure mathematics devoted to the study of the integers. German mathematician Carl Friedrich Gauss said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics." Number theorists study prime numbers as well as the properties of objects made out of integers or defined as generalizations of the integers. Integers can be considered either as solutions to equations. Questions in number theory are best understood through the study of analytical objects that encode properties of the integers, primes or other number-theoretic objects in some fashion. One may study real numbers in relation to rational numbers, for example, as approximated by the latter; the older term for number theory is arithmetic. By the early twentieth century, it had been superseded by "number theory"; the use of the term arithmetic for number theory regained some ground in the second half of the 20th century, arguably in part due to French influence. In particular, arithmetical is preferred as an adjective to number-theoretic.
The first historical find of an arithmetical nature is a fragment of a table: the broken clay tablet Plimpton 322 contains a list of "Pythagorean triples", that is, integers such that a 2 + b 2 = c 2. The triples are too large to have been obtained by brute force; the heading over the first column reads: "The takiltum of the diagonal, subtracted such that the width..." The table's layout suggests that it was constructed by means of what amounts, in modern language, to the identity 2 + 1 = 2, implicit in routine Old Babylonian exercises. If some other method was used, the triples were first constructed and reordered by c / a for actual use as a "table", for example, with a view to applications, it is not known whether there could have been any. It has been suggested instead. While Babylonian number theory—or what survives of Babylonian mathematics that can be called thus—consists of this single, striking fragment, Babylonian algebra was exceptionally well developed. Late Neoplatonic sources state.
Much earlier sources state that Pythagoras traveled and studied in Egypt. Euclid IX 21–34 is probably Pythagorean. Pythagorean mystics gave great importance to the even; the discovery that 2 is irrational is credited to the early Pythagoreans. By revealing that numbers could be irrational, this discovery seems to have provoked the first foundational crisis in mathematical history; this forced a distinction between numbers, on the one hand, lengths and proportions, on the other hand. The Pythagorean tradition spoke of so-called polygonal or figurate numbers. While square numbers, cubic numbers, etc. are seen now as more natural than triangular numbers, pentagonal numbers, etc. the study of the sums of triangular and pentagonal numbers would prove fruitful in the early modern period. We know of no arithmetical material in ancient Egyptian or Vedic sources, though there is some algebra in both; the Chinese remainder theorem appears as an exercise in Sunzi Suanjing There is some numerical mysticism in Chinese mathematics, unlike that of the Pythagoreans, it seems to have led nowhere.
Like the Pythagoreans' perfect numbers, magic squares have passed from superstition into recreation. Aside from a few fragments, the mathematics of Classical Greece is known to us either through the reports of contemporary non-m
A composite number is a positive integer that can be formed by multiplying two smaller positive integers. Equivalently, it is a positive integer that has at least one divisor other than itself; every positive integer is composite, prime, or the unit 1, so the composite numbers are the numbers that are not prime and not a unit. For example, the integer 14 is a composite number because it is the product of the two smaller integers 2 × 7; the integers 2 and 3 are not composite numbers because each of them can only be divided by one and itself. The composite numbers up to 150 are 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100, 102, 104, 105, 106, 108, 110, 111, 112, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 128, 129, 130, 132, 133, 134, 135, 136, 138, 140, 141, 142, 143, 144, 145, 146, 147, 148, 150.
Every composite number can be written as the product of two or more primes. For example, the composite number 299 can be written as 13 × 23, the composite number 360 can be written as 23 × 32 × 5; this fact is called the fundamental theorem of arithmetic. There are several known primality tests that can determine whether a number is prime or composite, without revealing the factorization of a composite input. One way to classify composite numbers is by counting the number of prime factors. A composite number with two prime factors is a 2-almost prime. A composite number with three distinct prime factors is a sphenic number. In some applications, it is necessary to differentiate between composite numbers with an odd number of distinct prime factors and those with an number of distinct prime factors. For the latter μ = 2 x = 1, while for the former μ = 2 x + 1 = − 1. However, for prime numbers, the function returns −1 and μ = 1. For a number n with one or more repeated prime factors, μ = 0. If all the prime factors of a number are repeated it is called a powerful number.
If none of its prime factors are repeated, it is called squarefree. For example, 72 = 23 × 32, all the prime factors are repeated, so 72 is a powerful number. 42 = 2 × 3 × 7, none of the prime factors are repeated, so 42 is squarefree. Another way to classify composite numbers is by counting the number of divisors. All composite numbers have at least three divisors. In the case of squares of primes, those divisors are. A number n that has more divisors than any x < n is a composite number. Composite numbers have been called "rectangular numbers", but that name can refer to the pronic numbers, numbers that are the product of two consecutive integers, yet another way to classify composite numbers is to determine whether all prime factors are either all below or all above some fixed number. Such numbers are called rough numbers, respectively. Canonical representation of a positive integer Integer factorization Sieve of Eratosthenes Table of prime factors Fraleigh, John B. A First Course In Abstract Algebra, Reading: Addison-Wesley, ISBN 0-201-01984-1 Herstein, I. N.
Topics In Algebra, Waltham: Blaisdell Publishing Company, ISBN 978-1114541016 Long, Calvin T. Elementary Introduction to Number Theory, Lexington: D. C. Heath and Company, LCCN 77-171950 McCoy, Neal H. Introduction To Modern Algebra, Revised Edition, Boston: Allyn and Bacon, LCCN 68-15225 Pettofrezzo, Anthony J.. Elements of Number Theory, Englewood Cliffs: Prentice Hall, LCCN 77-81766 Lists of composites with prime factorization Divisor Plot
A regular polyhedron is a polyhedron whose symmetry group acts transitively on its flags. A regular polyhedron is symmetrical, being all of edge-transitive, vertex-transitive and face-transitive. In classical contexts, many different equivalent definitions are used. A regular polyhedron is identified by its Schläfli symbol of the form, where n is the number of sides of each face and m the number of faces meeting at each vertex. There are 5 finite convex regular polyhedra, four regular star polyhedra, making nine regular polyhedra in all. In addition, there are five regular compounds of the regular polyhedra. There are five convex regular polyhedra, known as the Platonic solids, four regular star polyhedra, the Kepler–Poinsot polyhedra, five regular compounds of regular polyhedra: The property of having a similar arrangement of faces around each vertex can be replaced by any of the following equivalent conditions in the definition: The vertices of the polyhedron all lie on a sphere. All the dihedral angles of the polyhedron are equal All the vertex figures of the polyhedron are regular polygons.
All the solid angles of the polyhedron are congruent. A regular polyhedron has all of three related spheres which share its centre: An insphere, tangent to all faces. An intersphere or midsphere, tangent to all edges. A circumsphere, tangent to all vertices; the regular polyhedra are the most symmetrical of all the polyhedra. They lie in just three symmetry groups, which are named after them: Tetrahedral Octahedral Icosahedral Any shapes with icosahedral or octahedral symmetry will contain tetrahedral symmetry; the five Platonic solids have an Euler characteristic of 2. This reflects that the surface is a topological 2-sphere, so is true, for example,of any polyhedron, star-shaped with respect to some interior point; the sum of the distances from any point in the interior of a regular polyhedron to the sides is independent of the location of the point. However, the converse does not hold, not for tetrahedra. In a dual pair of polyhedra, the vertices of one polyhedron correspond to the faces of the other, vice versa.
The regular polyhedra show this duality as follows: The tetrahedron is self-dual, i.e. it pairs with itself. The cube and octahedron are dual to each other; the icosahedron and dodecahedron are dual to each other. The small stellated dodecahedron and great dodecahedron are dual to each other; the great stellated dodecahedron and great icosahedron are dual to each other. The Schläfli symbol of the dual is just the original written backwards. Stones carved in shapes resembling clusters of spheres or knobs have been found in Scotland and may be as much as 4,000 years old; some of these stones show not only the symmetries of the five Platonic solids, but some of the relations of duality amongst them. Examples of these stones are on display in the John Evans room of the Ashmolean Museum at Oxford University. Why these objects were made, or how their creators gained the inspiration for them, is a mystery. There is doubt regarding the mathematical interpretation of these objects, as many have non-platonic forms, only one has been found to be a true icosahedron, as opposed to a reinterpretation of the icosahedron dual, the dodecahedron.
It is possible that the Etruscans preceded the Greeks in their awareness of at least some of the regular polyhedra, as evidenced by the discovery near Padua in the late 19th century of a dodecahedron made of soapstone, dating back more than 2,500 years. The earliest known written records of the regular convex solids originated from Classical Greece; when these solids were all discovered and by whom is not known, but Theaetetus, was the first to give a mathematical description of all five. H. S. M. Coxeter credits Plato with having made models of them, mentions that one of the earlier Pythagoreans, Timaeus of Locri, used all five in a correspondence between the polyhedra and the nature of the universe as it was perceived – this correspondence is recorded in Plato's dialogue Timaeus. Euclid's reference to Plato led to their common description as the Platonic solids. One might characterise the Greek definition as follows: A regular polygon is a planar figure with all edges equal and all corners equal A regular polyhedron is a solid figure with all faces being congruent regular polygons, the same number arranged all alike around each vertex.
This definition rules out, for example, the square pyramid, or the shape formed by joining two tetrahedra together. This concept of a regular polyhedron would remain unchallenged for 2000 years. Regular star polygons such as the pentagram were known to the ancient Greeks – the pentagram was used by the Pythagoreans as their secret sign, but they did not use them to construct polyhedra, it was not until the early 17th century that Johannes Kepler realised that pentagrams could be used as the faces of regular star polyhedra. Some of these star polyhedra may have been discovered