What is a tiling?

... We could have weeded them out by requesting that not all l. Moreover, solution 8 appears in Figure 2(e) and solution 16 f). So we know that all these combinations are doable. That is to rk at one vertex, but they can be repeated again and again at oducing a mixed tiling of a plane. This is not the c ...

... We could have weeded them out by requesting that not all l. Moreover, solution 8 appears in Figure 2(e) and solution 16 f). So we know that all these combinations are doable. That is to rk at one vertex, but they can be repeated again and again at oducing a mixed tiling of a plane. This is not the c ...

COOL MATH! - James Tanton

... right at half the scale. Now repeat steps 1, 2, 3, and 4 for this smaller polygon to argue that it too must have even coordinates! Divide them each by two to produce another equilateral lattice polygon. By steps 1, 2, 3, and 4 its coordinates are even, and we can divide them each by two yet again. A ...

... right at half the scale. Now repeat steps 1, 2, 3, and 4 for this smaller polygon to argue that it too must have even coordinates! Divide them each by two to produce another equilateral lattice polygon. By steps 1, 2, 3, and 4 its coordinates are even, and we can divide them each by two yet again. A ...

Calculating generalised image and discriminant Milnor numbers in

... Assume that f : ðCn ; 0Þ ! ðCnþ1 ; 0Þ is a ﬁnitely A-determined corank 1 mapgerm, (we will abuse notation and also use f to denote a representative of the germ f ). Let ft : Ut ! Cnþ1 be a stabilisation of f, where Ut is an open domain in Cn . That is, ft is A-stable at every point. Definition 2.4. ...

... Assume that f : ðCn ; 0Þ ! ðCnþ1 ; 0Þ is a ﬁnitely A-determined corank 1 mapgerm, (we will abuse notation and also use f to denote a representative of the germ f ). Let ft : Ut ! Cnþ1 be a stabilisation of f, where Ut is an open domain in Cn . That is, ft is A-stable at every point. Definition 2.4. ...

Gabriel Lamé`s Counting of Triangulations

... In a 1751 letter to Christian Goldbach (1690–1764), Leonhard Euler (1707–1783) discusses the problem of counting the number of triangulations of a convex polygon. Euler, one of the most prolific mathematicians of all times, and Goldbach, who was a Professor of Mathematics and historian at St. Peters ...

... In a 1751 letter to Christian Goldbach (1690–1764), Leonhard Euler (1707–1783) discusses the problem of counting the number of triangulations of a convex polygon. Euler, one of the most prolific mathematicians of all times, and Goldbach, who was a Professor of Mathematics and historian at St. Peters ...

Document

... Theorem: Every elementary triangulation of a convex polygon with n vertices requires n – 3 lines. Proof: By complete induction. Let P(n) be “every elementary triangulation of a convex polygon requires n–3 lines.” We prove P(n) holds for all n ≥ 3. As a base case, we prove P(3): elementarily triangu ...

... Theorem: Every elementary triangulation of a convex polygon with n vertices requires n – 3 lines. Proof: By complete induction. Let P(n) be “every elementary triangulation of a convex polygon requires n–3 lines.” We prove P(n) holds for all n ≥ 3. As a base case, we prove P(3): elementarily triangu ...

Section 11.3 - Malibu High School

... pentagon ABCDE by using the formula A = —12 aP, or A = —12 a • ns. Then find the area by adding the areas of smaller polygons. Check that both methods yield the same area. Which method do you prefer? Explain ...

... pentagon ABCDE by using the formula A = —12 aP, or A = —12 a • ns. Then find the area by adding the areas of smaller polygons. Check that both methods yield the same area. Which method do you prefer? Explain ...

Math 2A: [10-6] Polygons

... 5. A square is a quadrilateral with 4 right angles and 4 congruent sides. ...

... 5. A square is a quadrilateral with 4 right angles and 4 congruent sides. ...

THE GENUS OF A QUADRATIC FORM Our basic problem is to

... We’re going to stay with the rational case, when there is only one such special prime, usually called “infinity.” However, we’ll go back to Hasse’s way of thinking about things, and call it “−1,” which is really much more natural. The “−1-adic” rational or integral numbers will be defined to be the ...

... We’re going to stay with the rational case, when there is only one such special prime, usually called “infinity.” However, we’ll go back to Hasse’s way of thinking about things, and call it “−1,” which is really much more natural. The “−1-adic” rational or integral numbers will be defined to be the ...

Modular Diagonal Quotient Surfaces (Survey)

... the (twisted) diagonal quotient surface defined by αε . Notes: 1) The modular diagonal quotient surfaces as defined above may be viewed as a special case of the general diagonal quotient surfaces studied in Kani/Schanz[7]. 2) The inclusions of subsgroups {1} ≤ ∆αε ≤ G × G induce finite morphisms (bo ...

... the (twisted) diagonal quotient surface defined by αε . Notes: 1) The modular diagonal quotient surfaces as defined above may be viewed as a special case of the general diagonal quotient surfaces studied in Kani/Schanz[7]. 2) The inclusions of subsgroups {1} ≤ ∆αε ≤ G × G induce finite morphisms (bo ...

Lesson 31-1: Sum of the Measures of the Interior Angles of Polygons

... Observe that as the number of sides increases by one, the sum of the angle measures also increases by a constant amount. What type of function models this behavior? linear ...

... Observe that as the number of sides increases by one, the sum of the angle measures also increases by a constant amount. What type of function models this behavior? linear ...

Topological conjugacy and symbolic dynamics

... such that |y − xN | ≤ 2−n . By the construction of x0 there exists an N such that the binary sequence for xN starts with b1 , b2 , . . . , bn . Thus |y − xN | ≤ 2−n as required. Remark There is a theorem (due to G.H.Hardy) that if an infinite sequence of 0’s and 1’s is chosen at random, then with ...

... such that |y − xN | ≤ 2−n . By the construction of x0 there exists an N such that the binary sequence for xN starts with b1 , b2 , . . . , bn . Thus |y − xN | ≤ 2−n as required. Remark There is a theorem (due to G.H.Hardy) that if an infinite sequence of 0’s and 1’s is chosen at random, then with ...

Math 4707 The Catalan Nunbers 1 Introduction

... Instead of counting the blockwalks directly, we will count the number of blockwalks which do cross the railroad tracks, and then subtract from the total number of blockwalks. Let B be a blockwalk which crosses the railroad track, and let Bi denote the letter (N or E) in the ith step of the walk. Any ...

... Instead of counting the blockwalks directly, we will count the number of blockwalks which do cross the railroad tracks, and then subtract from the total number of blockwalks. Let B be a blockwalk which crosses the railroad track, and let Bi denote the letter (N or E) in the ith step of the walk. Any ...

11.2 Practice with Examples

... The distance from the center to any side of a regular polygon is called the apothem of the polygon. A central angle of a regular polygon is an angle whose vertex is the center and whose sides contain two consecutive vertices of the polygon. Theorem 11.3 Area of an Equilateral Triangle The area of an ...

... The distance from the center to any side of a regular polygon is called the apothem of the polygon. A central angle of a regular polygon is an angle whose vertex is the center and whose sides contain two consecutive vertices of the polygon. Theorem 11.3 Area of an Equilateral Triangle The area of an ...

Fibonnaci Numbers, The Golden Ratio, and Platonic Solids

... each vertex, and the regular dodecahedron, wherein 3 pentagons meet at each vertex. These have 4, 8, 20, 6 and 12 sides respectively, and are named after the Greek words for these numbers (tetra- meaning four, octa- eight, etc). The regular means that all the polygons making up the polyhedron are co ...

... each vertex, and the regular dodecahedron, wherein 3 pentagons meet at each vertex. These have 4, 8, 20, 6 and 12 sides respectively, and are named after the Greek words for these numbers (tetra- meaning four, octa- eight, etc). The regular means that all the polygons making up the polyhedron are co ...

TILINGS 1. Set-up: Cut out regular polygons of equal size length and

... bees and suitable for their communal life. But why don’t the bees construct their beehives out of pentagonal, or octagonal shapes? In mathematical terms, we could pose this question as follows: Question 1: For which integer numbers n ≥ 3 does there exist a tiling of the plane by identical regular n- ...

... bees and suitable for their communal life. But why don’t the bees construct their beehives out of pentagonal, or octagonal shapes? In mathematical terms, we could pose this question as follows: Question 1: For which integer numbers n ≥ 3 does there exist a tiling of the plane by identical regular n- ...

Math 470 Spring `05

... c) the given formula and P(c) d) the given formula and P(x) (but x should be in parentheses) f) the given formula, (((y)P(z)) -> R(x,y)), ((y)P(z)), R(x,y), P(z) 5. a) no; y does not remain free b) yes. (Note that only the last occurrence of x is free.) c) yes , but nothing changes as x is not fr ...

... c) the given formula and P(c) d) the given formula and P(x) (but x should be in parentheses) f) the given formula, (((y)P(z)) -> R(x,y)), ((y)P(z)), R(x,y), P(z) 5. a) no; y does not remain free b) yes. (Note that only the last occurrence of x is free.) c) yes , but nothing changes as x is not fr ...

Regular and Semiregular Polyhedra

... The notion of regular polyhedron has been extended to the nonconvex case, resulting in four more Kepler-Poinsot solids. The definition has also been extended to higher dimensions. In four dimensions there are six regular convex polyhedra, and in every dimension thereafter there are only three. Semir ...

... The notion of regular polyhedron has been extended to the nonconvex case, resulting in four more Kepler-Poinsot solids. The definition has also been extended to higher dimensions. In four dimensions there are six regular convex polyhedra, and in every dimension thereafter there are only three. Semir ...

Homework 2

... F1 , and then show that if the statement is true for some Fk−1 and Fk , it is also true for Fk+1 ; then the result follows by induction! In this problem, we’ll prove the following fun fact: (∗) The only prime Fibonacci numbers are F4 and Fn when n > 4 is prime. Now at first this might seem very hard ...

... F1 , and then show that if the statement is true for some Fk−1 and Fk , it is also true for Fk+1 ; then the result follows by induction! In this problem, we’ll prove the following fun fact: (∗) The only prime Fibonacci numbers are F4 and Fn when n > 4 is prime. Now at first this might seem very hard ...

Edges - mathplease.com

... form a line ♥ Vertices – “corners” – the point where more than two polygons intersect. ...

... form a line ♥ Vertices – “corners” – the point where more than two polygons intersect. ...

In mathematics, the pentagram map is a discrete dynamical system on the moduli space of polygons in the projective plane. The pentagram map takes a given polygon, finds the intersections of the shortest diagonals of the polygon, and constructs a new polygon from these intersections.Richard Schwartz introduced the pentagram map for a general polygon in a 1992 paperthough it seems that the special case, in whichthe map is defined for pentagons only, goes back to an 1871 paper of Alfred Clebsch and a 1945 paper of Theodore Motzkin.The pentagram map is similar in spirit to the constructions underlying Desargues' Theorem and Poncelet's porism. It echoes the rationale and construction underlying a conjecture of Branko Grünbaum concerningthe diagonals of a polygon.