Patterned Triply Periodic Polyhedra
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Square Rectangle Triangle Diamond (Rhombus) Oval Cylinder Octagon Pentagon Cone Cube Hexagon Pyramid Sphere Star Circle
SQUARE RECTANGLE TRIANGLE DIAMOND (RHOMBUS) OVAL CYLINDER OCTAGON PENTAGON CONE CUBE HEXAGON PYRAMID SPHERE STAR CIRCLE Powered by: www.mymathtables.com Page 1 what is Rectangle? • A rectangle is a four-sided flat shape where every angle is a right angle (90°). means "right angle" and show equal sides. what is Triangle? • A triangle is a polygon with three edges and three vertices. what is Octagon? • An octagon (eight angles) is an eight-sided polygon or eight-gon. what is Hexagon? • a hexagon is a six-sided polygon or six-gon. The total of the internal angles of any hexagon is 720°. what is Pentagon? • a plane figure with five straight sides and five angles. what is Square? • a plane figure with four equal straight sides and four right angles. • every angle is a right angle (90°) means "right ang le" show equal sides. what is Rhombus? • is a flat shape with four equal straight sides. A rhombus looks like a diamond. All sides have equal length. Opposite sides are parallel, and opposite angles are equal what is Oval? • Many distinct curves are commonly called ovals or are said to have an "oval shape". • Generally, to be called an oval, a plane curve should resemble the outline of an egg or an ellipse. Powered by: www.mymathtables.com Page 2 What is Cube? • Six equal square faces.tweleve edges and eight vertices • the angle between two adjacent faces is ninety. what is Sphere? • no faces,sides,vertices • All points are located at the same distance from the center. what is Cylinder? • two circular faces that are congruent and parallel • faces connected by a curved surface. -
Applying the Polygon Angle
POLYGONS 8.1.1 – 8.1.5 After studying triangles and quadrilaterals, students now extend their study to all polygons. A polygon is a closed, two-dimensional figure made of three or more non- intersecting straight line segments connected end-to-end. Using the fact that the sum of the measures of the angles in a triangle is 180°, students learn a method to determine the sum of the measures of the interior angles of any polygon. Next they explore the sum of the measures of the exterior angles of a polygon. Finally they use the information about the angles of polygons along with their Triangle Toolkits to find the areas of regular polygons. See the Math Notes boxes in Lessons 8.1.1, 8.1.5, and 8.3.1. Example 1 4x + 7 3x + 1 x + 1 The figure at right is a hexagon. What is the sum of the measures of the interior angles of a hexagon? Explain how you know. Then write an equation and solve for x. 2x 3x – 5 5x – 4 One way to find the sum of the interior angles of the 9 hexagon is to divide the figure into triangles. There are 11 several different ways to do this, but keep in mind that we 8 are trying to add the interior angles at the vertices. One 6 12 way to divide the hexagon into triangles is to draw in all of 10 the diagonals from a single vertex, as shown at right. 7 Doing this forms four triangles, each with angle measures 5 4 3 1 summing to 180°. -
Regular Polyhedra of Index 2
REGULAR POLYHEDRA OF INDEX 2 A dissertation presented by Anthony Cutler to The Department of Mathematics In partial fulfillment of the requirements for the degree of Doctor of Philosophy in the field of Mathematics Northeastern University Boston, Massachusetts April, 2009 1 REGULAR POLYHEDRA OF INDEX 2 by Anthony Cutler ABSTRACT OF DISSERTATION Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics in the Graduate School of Arts and Sciences of Northeastern University, April 2009 2 We classify all finite regular polyhedra of index 2, as defined in Section 2 herein. The definition requires the polyhedra to be combinatorially flag transitive, but does not require them to have planar or convex faces or vertex-figures, and neither does it require the polyhedra to be orientable. We find there are 10 combinatorially regular polyhedra of index 2 with vertices on one orbit, and 22 infinite families of combinatorially regular polyhedra of index 2 with vertices on two orbits, where polyhedra in the same family differ only in the relative diameters of their vertex orbits. For each such polyhedron, or family of polyhedra, we provide the underlying map, as well as a geometric diagram showing a representative face for each face orbit, and a verification of the polyhedron’s combinatorial regularity. A self-contained completeness proof is given. Exactly five of the polyhedra have planar faces, which is consistent with a previously known result. We conclude by describing a non-Petrie duality relation among regular polyhedra of index 2, and suggest how it can be extended to other combinatorially regular polyhedra. -
Binary Icosahedral Group and 600-Cell
Article Binary Icosahedral Group and 600-Cell Jihyun Choi and Jae-Hyouk Lee * Department of Mathematics, Ewha Womans University 52, Ewhayeodae-gil, Seodaemun-gu, Seoul 03760, Korea; [email protected] * Correspondence: [email protected]; Tel.: +82-2-3277-3346 Received: 10 July 2018; Accepted: 26 July 2018; Published: 7 August 2018 Abstract: In this article, we have an explicit description of the binary isosahedral group as a 600-cell. We introduce a method to construct binary polyhedral groups as a subset of quaternions H via spin map of SO(3). In addition, we show that the binary icosahedral group in H is the set of vertices of a 600-cell by applying the Coxeter–Dynkin diagram of H4. Keywords: binary polyhedral group; icosahedron; dodecahedron; 600-cell MSC: 52B10, 52B11, 52B15 1. Introduction The classification of finite subgroups in SLn(C) derives attention from various research areas in mathematics. Especially when n = 2, it is related to McKay correspondence and ADE singularity theory [1]. The list of finite subgroups of SL2(C) consists of cyclic groups (Zn), binary dihedral groups corresponded to the symmetry group of regular 2n-gons, and binary polyhedral groups related to regular polyhedra. These are related to the classification of regular polyhedrons known as Platonic solids. There are five platonic solids (tetrahedron, cubic, octahedron, dodecahedron, icosahedron), but, as a regular polyhedron and its dual polyhedron are associated with the same symmetry groups, there are only three binary polyhedral groups(binary tetrahedral group 2T, binary octahedral group 2O, binary icosahedral group 2I) related to regular polyhedrons. -
Petrie Schemes
Canad. J. Math. Vol. 57 (4), 2005 pp. 844–870 Petrie Schemes Gordon Williams Abstract. Petrie polygons, especially as they arise in the study of regular polytopes and Coxeter groups, have been studied by geometers and group theorists since the early part of the twentieth century. An open question is the determination of which polyhedra possess Petrie polygons that are simple closed curves. The current work explores combinatorial structures in abstract polytopes, called Petrie schemes, that generalize the notion of a Petrie polygon. It is established that all of the regular convex polytopes and honeycombs in Euclidean spaces, as well as all of the Grunbaum–Dress¨ polyhedra, pos- sess Petrie schemes that are not self-intersecting and thus have Petrie polygons that are simple closed curves. Partial results are obtained for several other classes of less symmetric polytopes. 1 Introduction Historically, polyhedra have been conceived of either as closed surfaces (usually topo- logical spheres) made up of planar polygons joined edge to edge or as solids enclosed by such a surface. In recent times, mathematicians have considered polyhedra to be convex polytopes, simplicial spheres, or combinatorial structures such as abstract polytopes or incidence complexes. A Petrie polygon of a polyhedron is a sequence of edges of the polyhedron where any two consecutive elements of the sequence have a vertex and face in common, but no three consecutive edges share a commonface. For the regular polyhedra, the Petrie polygons form the equatorial skew polygons. Petrie polygons may be defined analogously for polytopes as well. Petrie polygons have been very useful in the study of polyhedra and polytopes, especially regular polytopes. -
Four-Dimensional Regular Polytopes
faculty of science mathematics and applied and engineering mathematics Four-dimensional regular polytopes Bachelor’s Project Mathematics November 2020 Student: S.H.E. Kamps First supervisor: Prof.dr. J. Top Second assessor: P. Kiliçer, PhD Abstract Since Ancient times, Mathematicians have been interested in the study of convex, regular poly- hedra and their beautiful symmetries. These five polyhedra are also known as the Platonic Solids. In the 19th century, the four-dimensional analogues of the Platonic solids were described mathe- matically, adding one regular polytope to the collection with no analogue regular polyhedron. This thesis describes the six convex, regular polytopes in four-dimensional Euclidean space. The focus lies on deriving information about their cells, faces, edges and vertices. Besides that, the symmetry groups of the polytopes are touched upon. To better understand the notions of regularity and sym- metry in four dimensions, our journey begins in three-dimensional space. In this light, the thesis also works out the details of a proof of prof. dr. J. Top, showing there exist exactly five convex, regular polyhedra in three-dimensional space. Keywords: Regular convex 4-polytopes, Platonic solids, symmetry groups Acknowledgements I would like to thank prof. dr. J. Top for supervising this thesis online and adapting to the cir- cumstances of Covid-19. I also want to thank him for his patience, and all his useful comments in and outside my LATEX-file. Also many thanks to my second supervisor, dr. P. Kılıçer. Furthermore, I would like to thank Jeanne for all her hospitality and kindness to welcome me in her home during the process of writing this thesis. -
Regular Polyhedra Through Time
Fields Institute I. Hubard Polytopes, Maps and their Symmetries September 2011 Regular polyhedra through time The greeks were the first to study the symmetries of polyhedra. Euclid, in his Elements showed that there are only five regular solids (that can be seen in Figure 1). In this context, a polyhe- dron is regular if all its polygons are regular and equal, and you can find the same number of them at each vertex. Figure 1: Platonic Solids. It is until 1619 that Kepler finds other two regular polyhedra: the great dodecahedron and the great icosahedron (on Figure 2. To do so, he allows \false" vertices and intersection of the (convex) faces of the polyhedra at points that are not vertices of the polyhedron, just as the I. Hubard Polytopes, Maps and their Symmetries Page 1 Figure 2: Kepler polyhedra. 1619. pentagram allows intersection of edges at points that are not vertices of the polygon. In this way, the vertex-figure of these two polyhedra are pentagrams (see Figure 3). Figure 3: A regular convex pentagon and a pentagram, also regular! In 1809 Poinsot re-discover Kepler's polyhedra, and discovers its duals: the small stellated dodecahedron and the great stellated dodecahedron (that are shown in Figure 4). The faces of such duals are pentagrams, and are organized on a \convex" way around each vertex. Figure 4: The other two Kepler-Poinsot polyhedra. 1809. A couple of years later Cauchy showed that these are the only four regular \star" polyhedra. We note that the convex hull of the great dodecahedron, great icosahedron and small stellated dodecahedron is the icosahedron, while the convex hull of the great stellated dodecahedron is the dodecahedron. -
Uniform Panoploid Tetracombs
Uniform Panoploid Tetracombs George Olshevsky TETRACOMB is a four-dimensional tessellation. In any tessellation, the honeycells, which are the n-dimensional polytopes that tessellate the space, Amust by definition adjoin precisely along their facets, that is, their ( n!1)- dimensional elements, so that each facet belongs to exactly two honeycells. In the case of tetracombs, the honeycells are four-dimensional polytopes, or polychora, and their facets are polyhedra. For a tessellation to be uniform, the honeycells must all be uniform polytopes, and the vertices must be transitive on the symmetry group of the tessellation. Loosely speaking, therefore, the vertices must be “surrounded all alike” by the honeycells that meet there. If a tessellation is such that every point of its space not on a boundary between honeycells lies in the interior of exactly one honeycell, then it is panoploid. If one or more points of the space not on a boundary between honeycells lie inside more than one honeycell, the tessellation is polyploid. Tessellations may also be constructed that have “holes,” that is, regions that lie inside none of the honeycells; such tessellations are called holeycombs. It is possible for a polyploid tessellation to also be a holeycomb, but not for a panoploid tessellation, which must fill the entire space exactly once. Polyploid tessellations are also called starcombs or star-tessellations. Holeycombs usually arise when (n!1)-dimensional tessellations are themselves permitted to be honeycells; these take up the otherwise free facets that bound the “holes,” so that all the facets continue to belong to two honeycells. In this essay, as per its title, we are concerned with just the uniform panoploid tetracombs. -
Right Triangles and the Pythagorean Theorem Related?
Activity Assess 9-6 EXPLORE & REASON Right Triangles and Consider △ ABC with altitude CD‾ as shown. the Pythagorean B Theorem D PearsonRealize.com A 45 C 5√2 I CAN… prove the Pythagorean Theorem using A. What is the area of △ ABC? Of △ACD? Explain your answers. similarity and establish the relationships in special right B. Find the lengths of AD‾ and AB‾ . triangles. C. Look for Relationships Divide the length of the hypotenuse of △ ABC VOCABULARY by the length of one of its sides. Divide the length of the hypotenuse of △ACD by the length of one of its sides. Make a conjecture that explains • Pythagorean triple the results. ESSENTIAL QUESTION How are similarity in right triangles and the Pythagorean Theorem related? Remember that the Pythagorean Theorem and its converse describe how the side lengths of right triangles are related. THEOREM 9-8 Pythagorean Theorem If a triangle is a right triangle, If... △ABC is a right triangle. then the sum of the squares of the B lengths of the legs is equal to the square of the length of the hypotenuse. c a A C b 2 2 2 PROOF: SEE EXAMPLE 1. Then... a + b = c THEOREM 9-9 Converse of the Pythagorean Theorem 2 2 2 If the sum of the squares of the If... a + b = c lengths of two sides of a triangle is B equal to the square of the length of the third side, then the triangle is a right triangle. c a A C b PROOF: SEE EXERCISE 17. Then... △ABC is a right triangle. -
Cyclic Quadrilaterals — the Big Picture Yufei Zhao [email protected]
Winter Camp 2009 Cyclic Quadrilaterals Yufei Zhao Cyclic Quadrilaterals | The Big Picture Yufei Zhao [email protected] An important skill of an olympiad geometer is being able to recognize known configurations. Indeed, many geometry problems are built on a few common themes. In this lecture, we will explore one such configuration. 1 What Do These Problems Have in Common? 1. (IMO 1985) A circle with center O passes through the vertices A and C of triangle ABC and intersects segments AB and BC again at distinct points K and N, respectively. The circumcircles of triangles ABC and KBN intersects at exactly two distinct points B and M. ◦ Prove that \OMB = 90 . B M N K O A C 2. (Russia 1995; Romanian TST 1996; Iran 1997) Consider a circle with diameter AB and center O, and let C and D be two points on this circle. The line CD meets the line AB at a point M satisfying MB < MA and MD < MC. Let K be the point of intersection (different from ◦ O) of the circumcircles of triangles AOC and DOB. Show that \MKO = 90 . C D K M A O B 3. (USA TST 2007) Triangle ABC is inscribed in circle !. The tangent lines to ! at B and C meet at T . Point S lies on ray BC such that AS ? AT . Points B1 and C1 lies on ray ST (with C1 in between B1 and S) such that B1T = BT = C1T . Prove that triangles ABC and AB1C1 are similar to each other. 1 Winter Camp 2009 Cyclic Quadrilaterals Yufei Zhao A B S C C1 B1 T Although these geometric configurations may seem very different at first sight, they are actually very related. -
Are Your Polyhedra the Same As My Polyhedra?
Are Your Polyhedra the Same as My Polyhedra? Branko Gr¨unbaum 1 Introduction “Polyhedron” means different things to different people. There is very little in common between the meaning of the word in topology and in geometry. But even if we confine attention to geometry of the 3-dimensional Euclidean space – as we shall do from now on – “polyhedron” can mean either a solid (as in “Platonic solids”, convex polyhedron, and other contexts), or a surface (such as the polyhedral models constructed from cardboard using “nets”, which were introduced by Albrecht D¨urer [17] in 1525, or, in a more mod- ern version, by Aleksandrov [1]), or the 1-dimensional complex consisting of points (“vertices”) and line-segments (“edges”) organized in a suitable way into polygons (“faces”) subject to certain restrictions (“skeletal polyhedra”, diagrams of which have been presented first by Luca Pacioli [44] in 1498 and attributed to Leonardo da Vinci). The last alternative is the least usual one – but it is close to what seems to be the most useful approach to the theory of general polyhedra. Indeed, it does not restrict faces to be planar, and it makes possible to retrieve the other characterizations in circumstances in which they reasonably apply: If the faces of a “surface” polyhedron are sim- ple polygons, in most cases the polyhedron is unambiguously determined by the boundary circuits of the faces. And if the polyhedron itself is without selfintersections, then the “solid” can be found from the faces. These reasons, as well as some others, seem to warrant the choice of our approach. -
Convex Polytopes and Tilings with Few Flag Orbits
Convex Polytopes and Tilings with Few Flag Orbits by Nicholas Matteo B.A. in Mathematics, Miami University M.A. in Mathematics, Miami University A dissertation submitted to The Faculty of the College of Science of Northeastern University in partial fulfillment of the requirements for the degree of Doctor of Philosophy April 14, 2015 Dissertation directed by Egon Schulte Professor of Mathematics Abstract of Dissertation The amount of symmetry possessed by a convex polytope, or a tiling by convex polytopes, is reflected by the number of orbits of its flags under the action of the Euclidean isometries preserving the polytope. The convex polytopes with only one flag orbit have been classified since the work of Schläfli in the 19th century. In this dissertation, convex polytopes with up to three flag orbits are classified. Two-orbit convex polytopes exist only in two or three dimensions, and the only ones whose combinatorial automorphism group is also two-orbit are the cuboctahedron, the icosidodecahedron, the rhombic dodecahedron, and the rhombic triacontahedron. Two-orbit face-to-face tilings by convex polytopes exist on E1, E2, and E3; the only ones which are also combinatorially two-orbit are the trihexagonal plane tiling, the rhombille plane tiling, the tetrahedral-octahedral honeycomb, and the rhombic dodecahedral honeycomb. Moreover, any combinatorially two-orbit convex polytope or tiling is isomorphic to one on the above list. Three-orbit convex polytopes exist in two through eight dimensions. There are infinitely many in three dimensions, including prisms over regular polygons, truncated Platonic solids, and their dual bipyramids and Kleetopes. There are infinitely many in four dimensions, comprising the rectified regular 4-polytopes, the p; p-duoprisms, the bitruncated 4-simplex, the bitruncated 24-cell, and their duals.