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Lesson Plans 1.1 © 2009 Zometool, Inc. Zometool is a registered trademark of Zometool Inc. • 1-888-966-3386 • www.zometool.com US Patents RE 33,785; 6,840,699 B2. Based on the 31-zone system, discovered by Steve Baer, Zomeworks Corp., USA www zometool ® com Zome System Table of Contents Builds Genius! Zome System in the Classroom Subjects Addressed by Zome System . .5 Organization of Plans . 5 Graphics . .5 Standards and Assessment . .6 Preparing for Class . .6 Discovery Learning with Zome System . .7 Cooperative vs Individual Learning . .7 Working with Diverse Student Groups . .7 Additional Resources . .8 Caring for your Zome System Kit . .8 Basic Concept Plans Geometric Shapes . .9 Geometry Is All Around Us . .11 2-D Polygons . .13 Animal Form . .15 Attention!…Angles . .17 Squares and Rectangles . .19 Try the Triangles . .21 Similar Triangles . .23 The Equilateral Triangle . .25 2-D and 3-D Shapes . .27 Shape and Number . .29 What Is Reflection Symmetry? . .33 Multiple Reflection Symmetries . .35 Rotational Symmetry . .39 What is Perimeter? . .41 Perimeter Puzzles . .43 What Is Area? . .45 Volume For Beginners . .47 Beginning Bubbles . .49 Rhombi . .53 What Are Quadrilaterals? . .55 Trying Tessellations . .57 Tilings with Quadrilaterals . .59 Triangle Tiles - I . .61 Translational Symmetries in Tilings . .63 Spinners . .65 Printing with Zome System . .67 © 2002 by Zometool, Inc. All rights reserved. 1 Table of Contents Zome System Builds Genius! Printing Cubes and Pyramids . .69 Picasso and Math . .73 Speed Lines! . .75 Squashing Shapes . .77 Cubes - I . .79 Cubes - II . .83 Cubes - III . .87 Cubes - IV . .89 Even and Odd Numbers . .91 Intermediate Concept Plans Descriptive Writing . .93 Bubbles II: Surface Tension and Minimal Surfaces . .95 Tallest Tower in the World . .99 The Livable City . .103 Prime Factors . .105 Plane Patterns . .109 Triangle Tiles - II . .111 Tilings with Multiple Symmetries . .115 Spiral Symmetry . .117 Symmetries in Quadrilateral Tilings . .119 Measures of Space - I: Length and Areas . .121 Measures of Space - II: Volumes . .125 3-D Triangles . .129 3-D Triangle Tiles . .133 Finding Tau . .137 Naming 2-D and 3-D Shapes . .141 Plato’s Solids - I . .145 Plato’s Solids - II . .149 2-D and 3-D Stars . .153 Kepler’s Tilings . .155 Richert-Penrose Tilings . .157 Bridge Building Project . ..

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Snub Cell Tetricosa
E COLE NORMALE SUPERIEURE ________________________ A Zo o of ` embeddable Polytopal Graphs Michel DEZA VP GRISHUHKIN LIENS ________________________ Département de Mathématiques et Informatique CNRS URA 1327 A Zo o of ` embeddable Polytopal Graphs Michel DEZA VP GRISHUHKIN LIENS January Lab oratoire dInformatique de lEcole Normale Superieure rue dUlm PARIS Cedex Tel Adresse electronique dmiensfr CEMI Russian Academy of Sciences Moscow A zo o of l embeddable p olytopal graphs MDeza CNRS Ecole Normale Superieure Paris VPGrishukhin CEMI Russian Academy of Sciences Moscow Abstract A simple graph G V E is called l graph if for some n 2 IN there 1 exists a vertexaddressing of each vertex v of G by a vertex av of the n cub e H preserving up to the scale the graph distance ie d v v n G d av av for all v 2 V We distinguish l graphs b etween skeletons H 1 n of a variety of well known classes of p olytop es semiregular regularfaced zonotop es Delaunay p olytop es of dimension and several generalizations of prisms and antiprisms Introduction Some notation and prop erties of p olytopal graphs and hypermetrics Vector representations of l metrics and hypermetrics 1 Regularfaced p olytop es Regular p olytop es Semiregular not regular p olytop es Regularfaced not semiregularp olytop es of dimension Prismatic graphs Moscow and Glob e graphs Stellated k gons Cup olas Antiwebs Capp ed antiprisms towers and fullerenes regularfaced not semiregular p olyhedra Zonotop es Delaunay p olytop es Small
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Arxiv:Math/9906062V1 [Math.MG] 10 Jun 1999 Udo Udmna Eerh(Rn 96-01-00166)
Embedding the graphs of regular tilings and star-honeycombs into the graphs of hypercubes and cubic lattices ∗ Michel DEZA CNRS and Ecole Normale Supérieure, Paris, France Mikhail SHTOGRIN Steklov Mathematical Institute, 117966 Moscow GSP-1, Russia Abstract We review the regular tilings of d-sphere, Euclidean d-space, hyperbolic d-space and Coxeter’s regular hyperbolic honeycombs (with infinite or star-shaped cells or vertex figures) with respect of possible embedding, isometric up to a scale, of their skeletons into a m-cube or m-dimensional cubic lattice. In section 2 the last remaining 2-dimensional case is decided: for any odd m ≥ 7, star-honeycombs m m {m, 2 } are embeddable while { 2 ,m} are not (unique case of non-embedding for dimension 2). As a spherical analogue of those honeycombs, we enumerate, in section 3, 36 Riemann surfaces representing all nine regular polyhedra on the sphere. In section 4, non-embeddability of all remaining star-honeycombs (on 3-sphere and hyperbolic 4-space) is proved. In the last section 5, all cases of embedding for dimension d> 2 are identified. Besides hyper-simplices and hyper-octahedra, they are exactly those with bipartite skeleton: hyper-cubes, cubic lattices and 8, 2, 1 tilings of hyperbolic 3-, 4-, 5-space (only two, {4, 3, 5} and {4, 3, 3, 5}, of those 11 have compact both, facets and vertex figures). 1 Introduction arXiv:math/9906062v1 [math.MG] 10 Jun 1999 We say that given tiling (or honeycomb) T has a l1-graph and embeds up to scale λ into m-cube Hm (or, if the graph is infinite, into cubic lattice Zm ), if there exists a mapping f of the vertex-set of the skeleton graph of T into the vertex-set of Hm (or Zm) such that λdT (vi, vj)= ||f(vi), f(vj)||l1 = X |fk(vi) − fk(vj)| for all vertices vi, vj, 1≤k≤m ∗This work was supported by the Volkswagen-Stiftung (RiP-program at Oberwolfach) and Russian fund of fundamental research (grant 96-01-00166).
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Four-Dimensional Regular Polytopes
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Tiling the Plane with Equilateral Convex Pentagons Maria Fischer1
Parabola Volume 52, Issue 3 (2016) Tiling the plane with equilateral convex pentagons Maria Fischer1 Mathematicians and non-mathematicians have been concerned with ﬁnding pentago- nal tilings for almost 100 years, yet tiling the plane with convex pentagons remains the only unsolved problem when it comes to monohedral polygonal tiling of the plane. One of the oldest and most well known pentagonal tilings is the Cairo tiling shown below. It can be found in the streets of Cairo, hence the name, and in many Islamic decorations. Figure 1: Cairo tiling There have been 15 types of such pentagons found so far, but it is not clear whether this is the complete list. We will look at the properties of these 15 types and then ﬁnd a complete list of equilateral convex pentagons which tile the plane – a problem which has been solved by Hirschhorn in 1983. 1Maria Fischer completed her Master of Mathematics at UNSW Australia in 2016. 1 Archimedean/Semi-regular tessellation An Archimedean or semi-regular tessellation is a regular tessellation of the plane by two or more convex regular polygons such that the same polygons in the same order surround each polygon. All of these polygons have the same side length. There are eight such tessellations in the plane: # 1 # 2 # 3 # 4 # 5 # 6 # 7 # 8 Number 5 and number 7 involve triangles and squares, number 1 and number 8 in- volve triangles and hexagons, number 2 involves squares and octagons, number 3 tri- angles and dodecagons, number 4 involves triangles, squares and hexagons and num- ber 6 squares, hexagons and dodecagons.
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ADDENDUM the Following Remarks Were Added in Proof (November 1966). Page 67. an Easy Modification of Exercise 4.8.25 Establishes
ADDENDUM The following remarks were added in proof (November 1966). Page 67. An easy modification of exercise 4.8.25 establishes the follow ing result of Wagner [I]: Every simplicial k-"complex" with at most 2 k ~ vertices has a representation in R + 1 such that all the "simplices" are geometric (rectilinear) simplices. Page 93. J. H. Conway (private communication) has established the validity of the conjecture mentioned in the second footnote. Page 126. For d = 2, the theorem of Derry [2] given in exercise 7.3.4 was found earlier by Bilinski [I]. Page 183. M. A. Perles (private communication) recently obtained an affirmative solution to Klee's problem mentioned at the end of section 10.1. Page 204. Regarding the question whether a(~) = 3 implies b(~) ~ 4, it should be noted that if one starts from a topological cell complex ~ with a(~) = 3 it is possible that ~ is not a complex (in our sense) at all (see exercise 11.1.7). On the other hand, G. Wegner pointed out (in a private communication to the author) that the 2-complex ~ discussed in the proof of theorem 11.1 .7 indeed satisfies b(~) = 4. Page 216. Halin's [1] result (theorem 11.3.3) has recently been genera lized by H. A. lung to all complete d-partite graphs. (Halin's result deals with the graph of the d-octahedron, i.e. the d-partite graph in which each class of nodes contains precisely two nodes.) The existence of the numbers n(k) follows from a recent result of Mader [1] ; Mader's result shows that n(k) ~ k.2(~) .
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Zome and Euler's Theorem Julia Robinson Day Hosted by Google Tom Davis [email protected] April 23, 2008
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*Cube Madness Fin
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