Geometry in Design Geometrical Construction in 3D Forms by Prof
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Final Poster
Associating Finite Groups with Dessins d’Enfants Luis Baeza, Edwin Baeza, Conner Lawrence, and Chenkai Wang Abstract Platonic Solids Rotation Group Dn: Regular Convex Polygon Approach Each finite, connected planar graph has an automorphism group G;such Following Magot and Zvonkin, reduce to easier cases using “hypermaps” permutations can be extended to automorphisms of the Riemann sphere φ : P1(C) P1(C), then composing β = φ f where S 2(R) P1(C). In 1984, Alexander Grothendieck, inspired by a result of f : 1( ) ! 1( )isaBely˘ımapasafunctionofeither◦ zn or ' P C P C Gennadi˘ıBely˘ıfrom 1979, constructed a finite, connected planar graph 4 zn/(zn +1)! 2 such that Aut(f ) Z or Aut(f ) D ,respectively. ' n ' n ∆β via certain rational functions β(z)=p(z)/q(z)bylookingatthe inverse image of the interval from 0 to 1. The automorphisms of such a Hypermaps: Rotation Group Zn graph can be identified with the Galois group Aut(β)oftheassociated 1 1 rational function β : P (C) P (C). In this project, we investigate how Rigid Rotations of the Platonic Solids I Wheel/Pyramids (J1, J2) ! w 3 (w +8) restrictive Grothendieck’s concept of a Dessin d’Enfant is in generating all n 2 I φ(w)= 1 1 z +1 64 (w 1) automorphisms of planar graphs. We discuss the rigid rotations of the We have an action : PSL2(C) P (C) P (C). β(z)= : v = n + n, e =2 n, f =2 − n ◦ ⇥ 2 !n 2 4 zn · Platonic solids (the tetrahedron, cube, octahedron, icosahedron, and I Zn = r r =1 and Dn = r, s s = r =(sr) =1 are the rigid I Cupola (J3, J4, J5) dodecahedron), the Archimedean solids, the Catalan solids, and the rotations of the regular convex polygons,with 4w 4(w 2 20w +105)3 I φ(w)= − ⌦ ↵ ⌦ 1 ↵ Rotation Group A4: Tetrahedron 3 2 Johnson solids via explicit Bely˘ımaps. -
Islamic Geometric Ornaments in Istanbul
►SKETCH 2 CONSTRUCTIONS OF REGULAR POLYGONS Regular polygons are the base elements for constructing the majority of Islamic geometric ornaments. Of course, in Islamic art there are geometric ornaments that may have different genesis, but those that can be created from regular polygons are the most frequently seen in Istanbul. We can also notice that many of the Islamic geometric ornaments can be recreated using rectangular grids like the ornament in our first example. Sometimes methods using rectangular grids are much simpler than those based or regular polygons. Therefore, we should not omit these methods. However, because methods for constructing geometric ornaments based on regular polygons are the most popular, we will spend relatively more time explor- ing them. Before, we start developing some concrete constructions it would be worthwhile to look into a few issues of a general nature. As we have no- ticed while developing construction of the ornament from the floor in the Sultan Ahmed Mosque, these constructions are not always simple, and in order to create them we need some knowledge of elementary geometry. On the other hand, computer programs for geometry or for computer graphics can give us a number of simpler ways to develop geometric fig- ures. Some of them may not require any knowledge of geometry. For ex- ample, we can create a regular polygon with any number of sides by rotat- ing a point around another point by using rotations 360/n degrees. This is a very simple task if we use a computer program and the only knowledge of geometry we need here is that the full angle is 360 degrees. -
Vertex-Transplants on a Convex Polyhedron
CCCG 2020, Saskatoon, Canada, August 5{7, 2020 Vertex-Transplants on a Convex Polyhedron Joseph O'Rourke Abstract Regular Tetrahedron. Let the four vertices of a regu- lar tetrahedron of unit edge length be v1; v2; v3 forming Given any convex polyhedron P of sufficiently many ver- the base, and apex v0. Place a point x on the v3v0 edge, tices n, and with no vertex's curvature greater than π, close to v3. Then one can form a digon starting from x it is possible to cut out a vertex, and paste the excised and surrounding v0 with geodesics γ1 and γ2 to a point y portion elsewhere along a vertex-to-vertex geodesic, cre- on 4v1v2v0, with jγ1j = jγ2j = 1. See Fig. 1(a,b). This ating a new convex polyhedron P0 of n + 2 vertices. digon can then be cut out and its hole sutured closed. The removed digon surface can be folded to a doubly covered triangle, and pasted into edge v1v2. The re- 1 Introduction sulting convex polyhedron guaranteed by Alexandrov's Theorem is a 6-vertex irregular octahedron P0. The goal of this paper is to prove the following theorem: Theorem 1 For any convex polyhedron P of n > N vertices, none of which have curvature greater than π, there is a vertex v0 that can be cut out along a digon of geodesics, and the excised surface glued to a geodesic on P connecting two vertices v1; v2. The result is a new convex polyhedron P0 with n + 2 vertices. N = 16 suffices. -
Polygon Review and Puzzlers in the Above, Those Are Names to the Polygons: Fill in the Blank Parts. Names: Number of Sides
Polygon review and puzzlers ÆReview to the classification of polygons: Is it a Polygon? Polygons are 2-dimensional shapes. They are made of straight lines, and the shape is "closed" (all the lines connect up). Polygon Not a Polygon Not a Polygon (straight sides) (has a curve) (open, not closed) Regular polygons have equal length sides and equal interior angles. Polygons are named according to their number of sides. Name of Degree of Degree of triangle total angles regular angles Triangle 180 60 In the above, those are names to the polygons: Quadrilateral 360 90 fill in the blank parts. Pentagon Hexagon Heptagon 900 129 Names: number of sides: Octagon Nonagon hendecagon, 11 dodecagon, _____________ Decagon 1440 144 tetradecagon, 13 hexadecagon, 15 Do you see a pattern in the calculation of the heptadecagon, _____________ total degree of angles of the polygon? octadecagon, _____________ --- (n -2) x 180° enneadecagon, _____________ icosagon 20 pentadecagon, _____________ These summation of angles rules, also apply to the irregular polygons, try it out yourself !!! A point where two or more straight lines meet. Corner. Example: a corner of a polygon (2D) or of a polyhedron (3D) as shown. The plural of vertex is "vertices” Test them out yourself, by drawing diagonals on the polygons. Here are some fun polygon riddles; could you come up with the answer? Geometry polygon riddles I: My first is in shape and also in space; My second is in line and also in place; My third is in point and also in line; My fourth in operation but not in sign; My fifth is in angle but not in degree; My sixth is in glide but not symmetry; Geometry polygon riddles II: I am a polygon all my angles have the same measure all my five sides have the same measure, what general shape am I? Geometry polygon riddles III: I am a polygon. -
Platonic and Archimedean Geometries in Multicomponent Elastic Membranes
Platonic and Archimedean geometries in multicomponent elastic membranes Graziano Vernizzia,1, Rastko Sknepneka, and Monica Olvera de la Cruza,b,c,2 aDepartment of Materials Science and Engineering, Northwestern University, Evanston, IL 60208; bDepartment of Chemical and Biological Engineering, Northwestern University, Evanston, IL 60208; and cDepartment of Chemistry, Northwestern University, Evanston, IL 60208 Edited by L. Mahadevan, Harvard University, Cambridge, MA, and accepted by the Editorial Board February 8, 2011 (received for review August 30, 2010) Large crystalline molecular shells, such as some viruses and fuller- possible to construct a lattice on the surface of a sphere such enes, buckle spontaneously into icosahedra. Meanwhile multi- that each site has only six neighbors. Consequently, any such crys- component microscopic shells buckle into various polyhedra, as talline lattice will contain defects. If one allows for fivefold observed in many organelles. Although elastic theory explains defects only then, according to the Euler theorem, the minimum one-component icosahedral faceting, the possibility of buckling number of defects is 12 fivefold disclinations. In the absence of into other polyhedra has not been explored. We show here that further defects (21), these 12 disclinations are positioned on the irregular and regular polyhedra, including some Archimedean and vertices of an inscribed icosahedron (22). Because of the presence Platonic polyhedra, arise spontaneously in elastic shells formed of disclinations, the ground state of the shell has a finite strain by more than one component. By formulating a generalized elastic that grows with the shell size. When γ > γà (γà ∼ 154) any flat model for inhomogeneous shells, we demonstrate that coas- fivefold disclination buckles into a conical shape (19). -
Formulas Involving Polygons - Lesson 7-3
you are here > Class Notes – Chapter 7 – Lesson 7-3 Formulas Involving Polygons - Lesson 7-3 Here’s today’s warmup…don’t forget to “phone home!” B Given: BD bisects ∠PBQ PD ⊥ PB QD ⊥ QB M Prove: BD is ⊥ bis. of PQ P Q D Statements Reasons Honors Geometry Notes Today, we started by learning how polygons are classified by their number of sides...you should already know a lot of these - just make sure to memorize the ones you don't know!! Sides Name 3 Triangle 4 Quadrilateral 5 Pentagon 6 Hexagon 7 Heptagon 8 Octagon 9 Nonagon 10 Decagon 11 Undecagon 12 Dodecagon 13 Tridecagon 14 Tetradecagon 15 Pentadecagon 16 Hexadecagon 17 Heptadecagon 18 Octadecagon 19 Enneadecagon 20 Icosagon n n-gon Baroody Page 2 of 6 Honors Geometry Notes Next, let’s look at the diagonals of polygons with different numbers of sides. By drawing as many diagonals as we could from one diagonal, you should be able to see a pattern...we can make n-2 triangles in a n-sided polygon. Given this information and the fact that the sum of the interior angles of a polygon is 180°, we can come up with a theorem that helps us to figure out the sum of the measures of the interior angles of any n-sided polygon! Baroody Page 3 of 6 Honors Geometry Notes Next, let’s look at exterior angles in a polygon. First, consider the exterior angles of a pentagon as shown below: Note that the sum of the exterior angles is 360°. -
Year 6 – Wednesday 24Th June 2020 – Maths
1 Year 6 – Wednesday 24th June 2020 – Maths Can I identify 3D shapes that have pairs of parallel or perpendicular edges? Parallel – edges that have the same distance continuously between them – parallel edges never meet. Perpendicular – when two edges or faces meet and create a 90o angle. 1 4. 7. 10. 2 5. 8. 11. 3. 6. 9. 12. C1 – Using the shapes above: 1. Name and sort the shapes into: a. Pyramids b. Prisms 2. Draw a table to identify how many faces, edges and vertices each shape has. 3. Write your own geometric definition: a. Prism b. pyramid 4. Which shape is the odd one out? Explain why. C2 – Using the shapes above; 1. Which of the shapes have pairs of parallel edges in: a. All their faces? b. More than one half of the faces? c. One face only? 2. The following shapes have pairs of perpendicular edges. Identify the faces they are in: 2 a. A cube b. A square based pyramid c. A triangular prism d. A cuboid 3. Which shape with straight edges has no perpendicular edges? 4. Which shape has perpendicular edges in the shape but not in any face? C3 – Using the above shapes: 1. How many faces have pairs of parallel edges in: a. A hexagonal pyramid? b. A decagonal (10-sided) based prism? c. A heptagonal based prism? d. Which shape has no face with parallel edges but has parallel edges in the shape? 2. How many faces have perpendicular edges in: a. A pentagonal pyramid b. A hexagonal pyramid c. -
Unit 6 Visualising Solid Shapes(Final)
• 3D shapes/objects are those which do not lie completely in a plane. • 3D objects have different views from different positions. • A solid is a polyhedron if it is made up of only polygonal faces, the faces meet at edges which are line segments and the edges meet at a point called vertex. • Euler’s formula for any polyhedron is, F + V – E = 2 Where F stands for number of faces, V for number of vertices and E for number of edges. • Types of polyhedrons: (a) Convex polyhedron A convex polyhedron is one in which all faces make it convex. e.g. (1) (2) (3) (4) 12/04/18 (1) and (2) are convex polyhedrons whereas (3) and (4) are non convex polyhedron. (b) Regular polyhedra or platonic solids: A polyhedron is regular if its faces are congruent regular polygons and the same number of faces meet at each vertex. For example, a cube is a platonic solid because all six of its faces are congruent squares. There are five such solids– tetrahedron, cube, octahedron, dodecahedron and icosahedron. e.g. • A prism is a polyhedron whose bottom and top faces (known as bases) are congruent polygons and faces known as lateral faces are parallelograms (when the side faces are rectangles, the shape is known as right prism). • A pyramid is a polyhedron whose base is a polygon and lateral faces are triangles. • A map depicts the location of a particular object/place in relation to other objects/places. The front, top and side of a figure are shown. Use centimetre cubes to build the figure. -
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. -
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. -
Locating Diametral Points
Results Math (2020) 75:68 Online First c 2020 Springer Nature Switzerland AG Results in Mathematics https://doi.org/10.1007/s00025-020-01193-5 Locating Diametral Points Jin-ichi Itoh, Costin Vˆılcu, Liping Yuan , and Tudor Zamfirescu Abstract. Let K be a convex body in Rd,withd =2, 3. We determine sharp sufficient conditions for a set E composed of 1, 2, or 3 points of bdK, to contain at least one endpoint of a diameter of K.Weextend this also to convex surfaces, with their intrinsic metric. Our conditions are upper bounds on the sum of the complete angles at the points in E. We also show that such criteria do not exist for n ≥ 4points. Mathematics Subject Classification. 52A10, 52A15, 53C45. Keywords. Convex body, diameter, geodesic diameter, diametral point. 1. Introduction The tangent cone at a point x in the boundary bdK of a convex body K can be defined using only neighborhoods of x in bdK. So, one doesn’t normally expect to get global information about K from the size of the tangent cones at one, two or three points. Nevertheless, in some cases this is what happens! A convex body K in Rd is a compact convex set with interior points; we shall consider only the cases d =2, 3. A convex surface in R3 is the boundary of a convex body in R3. Let S be a convex surface and x apointinS. Consider homothetic dilations of S with the centre at x and coefficients of homothety tending to infinity. The limit surface is called the tangent cone at x (see [1]), and is denoted by Tx. -
Towards Van Der Laan's Plastic Number in the Plane
Journal for Geometry and Graphics Volume 13 (2009), No. 2, 163–175. Towards van der Laan’s Plastic Number in the Plane Vera W. de Spinadel1, Antonia Redondo Buitrago2 1Laboratorio de Matem´atica y Dise˜no, Universidad de Buenos Aires Buenos Aires, Argentina email: vspinadel@fibertel.com.ar 2Departamento de Matem´atica, I.E.S. Bachiller Sabuco Avenida de Espa˜na 9, 02002 Albacete, Espa˜na email: [email protected] Abstract. In 1960 D.H. van der Laan, architect and member of the Benedic- tine order, introduced what he calls the “Plastic Number” ψ, as an ideal ratio for a geometric scale of spatial objects. It is the real solution of the cubic equation x3 x 1 = 0. This equation may be seen as example of a family of trinomials − − xn x 1=0, n =2, 3,... Considering the real positive roots of these equations we− define− these roots as members of a “Plastic Numbers Family” (PNF) com- prising the well known Golden Mean φ = 1, 618..., the most prominent member of the Metallic Means Family [12] and van der Laan’s Number ψ = 1, 324... Similar to the occurrence of φ in art and nature one can use ψ for defining special 2D- and 3D-objects (rectangles, trapezoids, ellipses, ovals, ovoids, spirals and even 3D-boxes) and look for natural representations of this special number. Laan’s Number ψ and the Golden Number φ are the only “Morphic Numbers” in the sense of Aarts et al. [1], who define such a number as the common solution of two somehow dual trinomials.