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TM Geometiles Building Ideas Patent Pending GeometilesTM is a product of TM www.geometiles.com Welcome to GeometilesTM! Here are some ideas of what you can build with your set. You can use them as a springboard for your imagination! Hints and instructions for making selected objects are in the back of this booklet. Platonic Solids CUBE CUBE 6 squares 12 isosceles triangles OCTAHEDRON OCTAHEDRON 8 equilateral triangles 16 scalene triangles 2 Building Ideas © 2015 Imathgination LLC REGULAR TETRAHEDRA 4 equilateral triangles; 8 scalene triangles; 16 equilateral triangles ICOSAHEDRON DODECAHEDRON 20 equilateral triangles 12 pentagons 3 Building Ideas © 2015 Imathgination LLC Selected Archimedean Solids CUBOCTAHEDRON ICOSIDODECAHEDRON 6 squares, 8 equilateral triangles 20 equilateral triangles, 12 pentagons Miscellaneous Solids DOUBLE TETRAHEDRON RHOMBIC PRISM 12 scalene triangles 8 scalene triangles; 4 rectangles 4 Building Ideas © 2015 Imathgination LLC PENTAGONAL ANTIPRISM HEXAGONAL ANTIPRISM 10 equilateral triangles, 2 pentagons. 24 equilateral triangles STELLA OCTANGULA, OR STELLATED OCTAHEDRON 24 equilateral triangles 5 Building Ideas © 2015 Imathgination LLC TRIRECTANGULAR TETRAHEDRON 12 isosceles triangles, 8 scalene triangles SCALENOHEDRON TRAPEZOHEDRON 8 scalene triangles 16 scalene triangles 6 Building Ideas © 2015 Imathgination LLC Playful shapes FLOWER 12 pentagons, 10 squares, 9 rectagles, 6 scalene triangles 7 Building Ideas © 2015 Imathgination LLC GEMSTONE 8 equilateral triangles, 8 rectangles, 4 isosceles triangles, 8 scalene triangles STAR 20 equilateral triangles, 2 pentagons 8 Building Ideas © 2015 Imathgination LLC CASTLE 6 equilateral triangles, 12 rectangles, 12 isosceles triangles, 6 scalene triangles, 4 squares 9 Building Ideas © 2015 Imathgination LLC CUBE WITH WINDOWS A.K.A. TISSUE BOX 24 scalene triangles 10 Building Ideas © 2015 Imathgination LLC Instructions and hints Small Octahedron 1 3 2 Large Octahedron 1 2 3 4 11 Building Ideas © 2015 Imathgination LLC Medium Sized Tetrahedron 1 2 4 3 12 Building Ideas © 2015 Imathgination LLC Icosahedron 1 2 3 4 6 5 13 Building Ideas © 2015 Imathgination LLC Dodecahedron 1 2 3 4 5 6 14 Building Ideas © 2015 Imathgination LLC Cuboctahedron 1 2 4 3 5 15 Building Ideas © 2015 Imathgination LLC Icosidodecahedron 1 2 3 4 5 6 7 16 Building Ideas © 2015 Imathgination LLC Stella Octangula 2 1 X 2 X 2 3 17 Building Ideas © 2015 Imathgination LLC Trirectangular Tetrahedron Bottom face Scalenohedron 2 1 Trapezohedron 2 1 18 Building Ideas © 2015 Imathgination LLC Flower Hint for making leaf for flower: first make the leaf using 4 scalene triangles. Right angle Then make the flower and attach leaf as shown with arrows: Leaf is attached Attach leaf here 19 Building Ideas © 2015 Imathgination LLC Star Attach pyramid last Cube with Windows 1 2 cover with lid 3 20 Building Ideas © 2015 Imathgination LLC .

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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.
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Computational Design Framework 3D Graphic Statics
Computational Design Framework for 3D Graphic Statics 3D Graphic for Computational Design Framework Computational Design Framework for 3D Graphic Statics Juney Lee Juney Lee Juney ETH Zurich • PhD Dissertation No. 25526 Diss. ETH No. 25526 DOI: 10.3929/ethz-b-000331210 Computational Design Framework for 3D Graphic Statics A thesis submitted to attain the degree of Doctor of Sciences of ETH Zurich (Dr. sc. ETH Zurich) presented by Juney Lee 2015 ITA Architecture & Technology Fellow Supervisor Prof. Dr. Philippe Block Technical supervisor Dr. Tom Van Mele Co-advisors Hon. D.Sc. William F. Baker Prof. Allan McRobie PhD defended on October 10th, 2018 Degree conﬁrmed at the Department Conference on December 5th, 2018 Printed in June, 2019 For my parents who made me, for Dahmi who raised me, and for Seung-Jin who completed me. Acknowledgements I am forever indebted to the Block Research Group, which is truly greater than the sum of its diverse and talented individuals. The camaraderie, respect and support that every member of the group has for one another were paramount to the completion of this dissertation. I sincerely thank the current and former members of the group who accompanied me through this journey from close and afar. I will cherish the friendships I have made within the group for the rest of my life. I am tremendously thankful to the two leaders of the Block Research Group, Prof. Dr. Philippe Block and Dr. Tom Van Mele. This dissertation would not have been possible without my advisor Prof. Block and his relentless enthusiasm, creative vision and inspiring mentorship.
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On a Remarkable Cube of Pyrite, Carrying Crys- Tallized Gold and Galena of Unusual Habit
ON A REMARKABLE CUBE OF PYRITE, CARRYING CRYS- TALLIZED GOLD AND GALENA OF UNUSUAL HABIT By JOSEPH E. POGUE Assistant Curator, Division of Mineralogy, U. S. National Museum With One Plate The intergrowth or interpenetration of two or more minerals, especially if these be well crystallized, often shows a certain mutual crystallographic control in the arrangement of the individuals, sug- gestive of interacting molecular forces. Occasionally a crystal upon nearly completing its growth exerts what may be termed "surface affinit}'," in that it seems to attract molecules of composition differ- ent from its own and causes these to crystallize in positions bearing definite crystallographic relations to the host crystal, as evidenced, for example, by the regular arrangement of marcasite on calcite, chalcopyrite on galena, quartz on fluorite, and so on. Of special interest, not only because exhibiting the features mentioned above, but also on account of the unusual development of the individuals and the great beauty of the specimen, is a large cube of pyrite, studded with crystals of native gold and partly covered by plates of galena, acquired some years ago by the U. S. National Museum. This cube measures about 2 inches (51 mm.) along its edge, and is prominently striated, as is often the case with pyrite. It contains something more than 130 crystals of gold attached to its surface, has about one-fourth of its area covered with galena, and upon one face shows an imperfect crystal of chalcopyrite. The specimen came into the possession of the National Museum in 1906 and was ob- tained from the Snettisham District, near Juneau, Southeast Alaska.
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Framing Cyclic Revolutionary Emergence of Opposing Symbols of Identity Eppur Si Muove: Biomimetic Embedding of N-Tuple Helices in Spherical Polyhedra - /
Alternative view of segmented documents via Kairos 23 October 2017 | Draft Framing Cyclic Revolutionary Emergence of Opposing Symbols of Identity Eppur si muove: Biomimetic embedding of N-tuple helices in spherical polyhedra - / - Introduction Symbolic stars vs Strategic pillars; Polyhedra vs Helices; Logic vs Comprehension? Dynamic bonding patterns in n-tuple helices engendering n-fold rotating symbols Embedding the triple helix in a spherical octahedron Embedding the quadruple helix in a spherical cube Embedding the quintuple helix in a spherical dodecahedron and a Pentagramma Mirificum Embedding six-fold, eight-fold and ten-fold helices in appropriately encircled polyhedra Embedding twelve-fold, eleven-fold, nine-fold and seven-fold helices in appropriately encircled polyhedra Neglected recognition of logical patterns -- especially of opposition Dynamic relationship between polyhedra engendered by circles -- variously implying forms of unity Symbol rotation as dynamic essential to engaging with value-inversion References Introduction The contrast to the geocentric model of the solar system was framed by the Italian mathematician, physicist and philosopher Galileo Galilei (1564-1642). His much-cited phrase, " And yet it moves" (E pur si muove or Eppur si muove) was allegedly pronounced in 1633 when he was forced to recant his claims that the Earth moves around the immovable Sun rather than the converse -- known as the Galileo affair. Such a shift in perspective might usefully inspire the recognition that the stasis attributed so widely to logos and other much-valued cultural and heraldic symbols obscures the manner in which they imply a fundamental cognitive dynamic. Cultural symbols fundamental to the identity of a group might then be understood as variously moving and transforming in ways which currently elude comprehension.
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Shape Skeletons Creating Polyhedra with Straws
Shape Skeletons Creating Polyhedra with Straws Topics: 3-Dimensional Shapes, Regular Solids, Geometry Materials List Drinking straws or stir straws, cut in Use simple materials to investigate regular or advanced 3-dimensional shapes. half Fun to create, these shapes make wonderful showpieces and learning tools! Paperclips to use with the drinking Assembly straws or chenille 1. Choose which shape to construct. Note: the 4-sided tetrahedron, 8-sided stems to use with octahedron, and 20-sided icosahedron have triangular faces and will form sturdier the stir straws skeletal shapes. The 6-sided cube with square faces and the 12-sided Scissors dodecahedron with pentagonal faces will be less sturdy. See the Taking it Appropriate tool for Further section. cutting the wire in the chenille stems, Platonic Solids if used This activity can be used to teach: Common Core Math Tetrahedron Cube Octahedron Dodecahedron Icosahedron Standards: Angles and volume Polyhedron Faces Shape of Face Edges Vertices and measurement Tetrahedron 4 Triangles 6 4 (Measurement & Cube 6 Squares 12 8 Data, Grade 4, 5, 6, & Octahedron 8 Triangles 12 6 7; Grade 5, 3, 4, & 5) Dodecahedron 12 Pentagons 30 20 2-Dimensional and 3- Dimensional Shapes Icosahedron 20 Triangles 30 12 (Geometry, Grades 2- 12) 2. Use the table and images above to construct the selected shape by creating one or Problem Solving and more face shapes and then add straws or join shapes at each of the vertices: Reasoning a. For drinking straws and paperclips: Bend the (Mathematical paperclips so that the 2 loops form a “V” or “L” Practices Grades 2- shape as needed, widen the narrower loop and insert 12) one loop into the end of one straw half, and the other loop into another straw half.
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Relationship Between the Mandelbrot Algorithm and the Platonic Solids
Relationship between the Mandelbrot Algorithm and the Platonic Solids André Vallières∗ and Dominic Rochon† Département de mathématiques et d’informatique, Université du Québec C.P. 500, Trois-Rivières, Québec, Canada, G9A 5H7. « Chaque flot est un ondin qui nage dans le courant, chaque courant est un sentier qui serpente vers mon palais, et mon palais est bâti fluide, au fond du lac, dans le triangle du feu, de la terre et de l’air. » — Aloysius Bertrand, Gaspard de la nuit, 1842 Abstract This paper focuses on the dynamics of the eight tridimensional princi- pal slices of the tricomplex Mandelbrot set: the Tetrabrot, the Arrowhead- brot, the Mousebrot, the Turtlebrot, the Hourglassbrot, the Metabrot, the Airbrot (octahedron) and the Firebrot (tetrahedron). In particular, we establish a geometrical classification of these 3D slices using the proper- ties of some specific sets that correspond to projections of the bicomplex Mandelbrot set on various two-dimensional vector subspaces, and we prove that the Firebrot is a regular tetrahedron. Finally, we construct the so- called “Stella octangula” as a tricomplex dynamical system composed of the union of the Firebrot and its dual, and after defining the idempotent 3D slices of M3, we show that one of them corresponds to a third Platonic solid: the cube. AMS subject classification: 32A30, 30G35, 00A69, 51M20 Keywords: Generalized Mandelbrot Sets, Tricomplex Dynamics, Metatron- arXiv:2107.04016v3 [math.DS] 8 Sep 2021 brot, 3D Fractals, Platonic Solids, Airbrot, Earthbrot, Firebrot, Stella Octan- gula Introduction Quadratic polynomials iterated on hypercomplex algebras have been used to generate multidimensional Mandelbrot sets for several years [3,9, 11, 13, 17, ∗E-mail: [email protected] †E-mail: [email protected] 1 23, 28, 34].
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Wholemovement of the Circle Bradford Hansen-Smith 4606 N
Wholemovement of the circle Bradford Hansen-Smith 4606 N. Elston #3, Chicago IL 60630, USA [email protected] Wholeness is the most practical context in which to process information. The circle is Whole. It is a comprehensive tool for modeling patterns of forms and spatial organization inherent in our universe. Folding paper plates circles demonstrates the concept and process of movement within the Whole. Wholemovement generates individualized expressions of endless differences within the singularity of the circle. How we process information is determined by cultural conditions. Our educational system values past experiences of selected groups of people over present individual experience. That is how we define ourselves. We learn past processes used to solve past problems. This decreases our ability to see meaningful connections within a greater context. Connecting across diverse disciplines, bridging cultural and individual differences is a problem when viewed as separated pieces needing to be connected. Our condition of mind is a construct that supports methods of processing information by separating pieces. A greater understanding is emerging showing only the interactions of endless connections of extraordinarily diverse relationships, all principled to the movement of the Whole. We accept the circle as image. This is not questioned. We draw pictures of circles, traditionally using parts to construct fragmented symbols to explain 2-D and 3-D geometry and other mathematically related concepts. These image symbols are important in the development of mathematics. The symbol is the first step in abstracting information from its spatial context, allowing for greater manipulation of parts. By constructing images and using logic to find connections, we piece symbols together, looking to find some kind of greater meaning.
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PMA Day Workshop
PMA Day Workshop March 25, 2017 Presenters: Jeanette McLeod Phil Wilson Sarah Mark Nicolette Rattenbury 1 Contents The Maths Craft Mission ...................................................................................................................... 4 About Maths Craft ............................................................................................................................ 6 Our Sponsors .................................................................................................................................... 7 The Möbius Strip .................................................................................................................................. 8 The Mathematics of the Möbius .................................................................................................... 10 One Edge, One Side .................................................................................................................... 10 Meet the Family ......................................................................................................................... 11 Making and Manipulating a Möbius Strip ...................................................................................... 13 Making a Möbius ........................................................................................................................ 13 Manipulating a Möbius .............................................................................................................. 13 The Heart of a Möbius ..............................................................................................................
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7 Dee's Decad of Shapes and Plato's Number.Pdf
Dee’s Decad of Shapes and Plato’s Number i © 2010 by Jim Egan. All Rights reserved. ISBN_10: ISBN-13: LCCN: Published by Cosmopolite Press 153 Mill Street Newport, Rhode Island 02840 Visit johndeetower.com for more information. Printed in the United States of America ii Dee’s Decad of Shapes and Plato’s Number by Jim Egan Cosmopolite Press Newport, Rhode Island C S O S S E M R O P POLITE “Citizen of the World” (Cosmopolite, is a word coined by John Dee, from the Greek words cosmos meaning “world” and politês meaning ”citizen”) iii Dedication To Plato for his pursuit of “Truth, Goodness, and Beauty” and for writing a mathematical riddle for Dee and me to figure out. iv Table of Contents page 1 “Intertransformability” of the 5 Platonic Solids 15 The hidden geometric solids on the Title page of the Monas Hieroglyphica 65 Renewed enthusiasm for the Platonic and Archimedean solids in the Renaissance 87 Brief Biography of Plato 91 Plato’s Number(s) in Republic 8:546 101 An even closer look at Plato’s Number(s) in Republic 8:546 129 Plato shows his love of 360, 2520, and 12-ness in the Ideal City of “The Laws” 153 Dee plants more clues about Plato’s Number v vi “Intertransformability” of the 5 Platonic Solids Of all the polyhedra, only 5 have the stuff required to be considered “regular polyhedra” or Platonic solids: Rule 1. The faces must be all the same shape and be “regular” polygons (all the polygon’s angles must be identical).
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Stellated Cuboctahedron Through Post-Stellation of an M<Sub>
ARTICLES PUBLISHED ONLINE: 11 MARCH 2012 | DOI: 10.1038/NCHEM.1285 An M18L24 stellated cuboctahedron through post-stellation of an M12L24 core Qing-Fu Sun, Sota Sato and Makoto Fujita* Platonic and Archimedean polyhedra, well-known to mathematicians, have been recently constructed by chemists at a molecular scale by deﬁning the vertices and the edges with metal ions (M) and organic ligands (L), respectively. Here, we report the ﬁrst synthesis of a concave-surface ‘stellated polyhedron’, constructed by extending the faces of its precursor polyhedron until they intersect, forming additional nodes. Our approach involves the formation of an M12L24 cuboctahedron core, the linkers of which each bear a pendant ligand site that is subsequently able to bind an additional metal centre to form the stellated M18L24 cuboctahedron. During this post-stellation process, the square faces of the M12L24 core are closed by coordination of the pendant moieties to the additional metal centres, but they are re-opened on removing these metals ions from the vertices. This behaviour is reminiscent of the analogous metal-triggered gate opening–closing switches found in spherical virus capsids. tellated polyhedra1 are a unique family of polyhedra with resonance (NMR) spectroscopy and mass spectrometry (MS). The concave surfaces, constructed by extending the faces of a large square windows (Fig. 1) of the M12L24 cuboctahedron are Spolyhedron until they intersect (Fig. 1). Although the closed by the stellation, but can be re-opened by removal of the molecular self-assembly of polyhedral architectures is a topic of metal ions from the vertices, thereby providing a reversible gate current interest2–18, their stellated derivatives—in which the stellated opening–closing function for a large cuboctahedral cage.
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Appendix H: Missouri MAA Papers
1 Appendix H: Missouri MAA Papers 1916 (St. Louis Central High School) 1. A Course for Juniors in the School of Education. Professor L. D. Ames, University of Missouri. 2. Graphical Solution of Spherical Triangles. Professor W. H. Roever, Washington University 3. The Place of the Calculus in the Training of the High School Teacher. Professor Byron Cosby, Kirksville Normal School. 4. A Geometric Treatment of the Exponential Function. Dr. Otto Dunkel, Washington University. 5. Formulas for Approximate Integration. Professor Byron Ingold, Christian University 6. Claims of Mathematics in the High School Course of Study. Mr. H. P. Stellwagen, Yeatman High School 7. An Illustration of a Certain Necessary Condition in Minimizing a Definite Integral with a Discontinuous Integrand. Dr. Paul R. Rider, Washington University 8. On a Method of Sectioning Freshman and Sophomore Classes in Mathematics. Mr. Alan Campbell, Washington University. 9. The Equations and Models of a Large Group of Warped Surfaces. Professor C. A. Waldo, Washington University. 1917 (Kansas City Public Library) 1. Some Properties of Plane and Spherical Triangles and their Frequent Analogies. Professor William H. Zeigel, Missouri State Normal School, Kirksville. 2. The Value of Mathematics in Secondary Education. Dr. John W. Withers, Superintendent of Instruction, St. Louis 3. Sundials and Skylights. Professor William H. Roever, Washington University, St. Louis 4. Pure and Applied Mathematics in the Nineteenth Century. Professor G. R. Dean, Missouri School of Mines, Rolla. 5. The Equal Parallax Curve for Frontal and Lateral Vision. Dr. Paul R. Rider, Washington University, St. Louis. 6. A Simple Derivation of the Derivatives of the Trigonometric Functions.
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Folding the Circle As Both Whole and Part
BRIDGES Mathematical Connections in Art, Music, and Science Folding the Circle as Both Whole and Part Bradford Hansen-Smith 4606 N. Elston #3 Chicago IL 60630, USA [email protected] Abstract This paper is an introduction to fourteen years of proportional folding paper plates and observing the information- generating and modeling capabilities of the circle. The process of folding circles expands our understanding about the nature of the circle that is not predictable by drawing circles. Unlike all other shapes and forms the circle functions as both Whole and part at the same time. Folding and joining circles is a transformational process generating patterned formations and systems that closely reflects what is observed in nature. It is possible that the circle is the most comprehensive and inclusive hands-on tool we have for modeling patterns, generating forms, revealing abstractions of function, and principles that are common to the arts, mathematics, and the sciences. Introduction The circle is much more than what we call it. Draw a circle. Cut out around the picture of the circle. You have in your hand a circle in space. This paper will explore the process of folding the paper circle. The circle is about movement patterns, and how the circle shape is reformed into traditional polyhedra and many non-traditional polyhedral forms and systems. There are examples of how, by joining multiples of circle-folded forms, systems of great complexity and diversity of forms can be generated. The focus is on the folding process that allows the creased circle to form and be reformed, changing the arrangements of the circle surface, forming interval spaces, and shifting between symmetries.
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