Tessellating 'Honeycombs'
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Cosmic Architecture Kozmička Arhitektura
je značio nešto poput ‘univerzuma, reda i ornamenta’. [...] Za stare Grke riječ ‘Kosmos’ stavljena je u opreku s riječi ‘Chaos’. Kaos je prethodio nastanku svijeta kakvog poznajemo, ali ga Kozmička je naslijedio Kozmos koji je simbolizirao apsolutni red svi- jeta i ukupnost njegovih prirodnih fenomena. [...] Stari grčki ‘Kosmeo’ znači ‘rasporediti, urediti i ukrasiti’, a osoba Kosmése arhitektura (ukrašava) sebe kako bi svoj Kozmos učinila vidljivim.2 kako ¶ Ornament ima gramatiku. Ornament bi trebao posje- dovati prikladnost, proporcije, sklad čiji je rezultat mir... onaj mir koji um osjeća kada su oko, intelekt i naklonosti zadovo- Cosmic ljeni.3 ¶ Vjerujem, kao što sam rekao, da se može projektirati izvrsna i lijepa zgrada koju neće krasiti nikakvi ornamenti; ali jednako čvrsto vjerujem da se ukrašenu građevinu, skladno zamišljenu, dobro promišljenu, ne može lišiti njezinog sustava architecture ornamenata, a da se ne uništi njezina individualnost.4 ¶ Tipičan postupak drevne arhitekture je dodavanje idealnih aspekata ili idealnih struktura površini zgrade. [...] Cijepanje ili klizanje stvarne površine zida u izražajnu površinu je čin transfor- macije. ¶ Govori li nepravilna evolucija kamena o nevjerojat- noj gotičkoj priči o ljudskom životu? Ili je to usputna pojava nevažnih činjenica iscrpljenih kamenoloma i klesara? Ili je to pustolovina vremena? 5 gdje ¶ Ornament je svjesna zanatska intervencija u proi- napisao fotografije Arhiva / Archive Alberto Alessi (aaa) zvodnji polugotovih proizvoda, prije nego što budu montirani written by photographs by Arhiva / Archive Alinari (aa) na gradilištu. Ornament stvara sidrenu točku protiv homo- Ruskin Library, University of Lancaster (rl) genizacije i uniformnosti suvremene građevinske produkcije. Ornamentacija omogućuje izravan odgovor na lokalne uvjete proizvodnje, na geografske ili kulturne osobitosti. -
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. -
On the Application of the Honeycomb Conjecture to the Bee's Honeycomb
On the Application of the Honeycomb Conjecture to the Bee’s Honeycomb∗y Tim Räzz July 29, 2014 Abstract In a recent paper, Aidan Lyon and Mark Colyvan have proposed an explanation of the structure of the bee’s honeycomb based on the mathematical Honeycomb Conjecture. This explanation has instantly become one of the standard examples in the philosophical debate on mathematical explanations of physical phenomena. In this critical note, I argue that the explanation is not scientifically adequate. The reason for this is that the explanation fails to do justice to the essen- tially three-dimensional structure of the bee’s honeycomb. Contents 1 Introduction 2 2 Lyon’s and Colyvan’s Explanation 2 3 Baker: A Philosophical Motivation 3 4 Why the Explanation Fails 4 4.1 The Explanation is Incomplete . .4 4.2 The Honeycomb Conjecture Is (Probably) Irrelevant . .7 ∗Acknowledgements: I am grateful to Mark Colyvan, Matthias Egg, Michael Esfeld, Martin Gasser, Marion Hämmerli, Michael Messerli, Antoine Muller, Christian Sachse, Tilman Sauer, Raphael Scholl, two anonymous referees, and the participants of the phi- losophy of science research seminar in the fall of 2012 in Lausanne for discussions and comments on previous drafts of this paper. The usual disclaimer applies. This work was supported by the Swiss National Science Foundation, grants (100011-124462/1), (100018- 140201/1) yThis is a pre-copyedited, author-produced PDF of an article accepted for publication in Philosophia Mathematica following peer review. The definitive publisher-authenticated version, Räz, T. (2013): On the Application of the Honeycomb Conjecture to the Bee’s Honeycomb. -
Theorizing Ornament Estelle Thibault
From Herbal to Grammar : Theorizing Ornament Estelle Thibault To cite this version: Estelle Thibault. From Herbal to Grammar : Theorizing Ornament. Fourth International Conference of the European Architectural History Network, Jun 2016, Dublin, Ireland. pp.384-394. hal-01635839 HAL Id: hal-01635839 https://hal.archives-ouvertes.fr/hal-01635839 Submitted on 27 Oct 2019 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. EAHN Dublin 2016 1 PROCEEDINGS OF THE FOURTH INTERNATIONAL CONFERENCE OF THE EUROPEAN ARCHITECTURAL HISTORY NETWORK Edited by Kathleen James-Chakraborty EAHN Dublin 2016 2 Published by UCD School of Art History and Cultural Policy University College Dublin, Belfield, Dublin, Ireland. Copyright © UCD School of Art History and Cultural Policy No images in this publication may be reproduced without permission of the copyright holder. ISBN 978-1-5262-0376-2 EAHN Dublin 2016 3 * Indicates full paper included Table of Contents KEYNOTE .................................................................................................................................... -
Designing Modular Sculpture Systems
Bridges 2017 Conference Proceedings Designing Modular Sculpture Systems Christopher Carlson Wolfram Research, Inc 100 Trade Centre Drive Champaign, IL, 61820, USA E-mail: [email protected] Abstract We are interested in the sculptural possibilities of closed chains of modular units. One such system is embodied in MathMaker, a set of wooden pieces that can be connected end-to-end to create a fascinating variety of forms. MathMaker is based on a cubic honeycomb. We explore the possibilities of similar systems based on octahedral- tetrahedral, rhombic dodecahedral, and truncated octahedral honeycombs. Introduction The MathMaker construction kit consists of wooden parallelepipeds that can be connected end-to-end to make a great variety of sculptural forms [1]. Figure 1 shows on the left an untitled sculpture by Koos Verhoeff based on that system of modular units [2]. MathMaker derives from a cubic honeycomb. Each unit connects the center of one cubic face to the center of an adjacent face (Figure 1, center). Units connect via square planar faces which echo the square faces of cubes in the honeycomb. The square connecting faces can be matched in 4 ways via 90° rotation, so from any position each adjacent cubic face is accessible. A chain of MathMaker units jumps from cube to adjacent cube, adjacent units never traversing the same cube. A variant of MathMaker called “turned cross mitre” has units whose square cross-sections are rotated 45° about their central axes relative to the standard MathMaker units (Figure 1, right). Although the underlying cubic honeycomb structure is the same, the change in unit yields structures with strikingly different visual impressions. -
Polyhedra and Packings from Hyperbolic Honeycombs
Polyhedra and packings from hyperbolic honeycombs Martin Cramer Pedersena,1 and Stephen T. Hydea aDepartment of Applied Mathematics, Research School of Physics and Engineering, Australian National University, Canberra ACT 2601, Australia Edited by Robion C. Kirby, University of California, Berkeley, CA, and approved May 25, 2018 (received for review November 29, 2017) We derive more than 80 embeddings of 2D hyperbolic honey- Disc Packings and Triangular Patterns combs in Euclidean 3 space, forming 3-periodic infinite polyhedra The density of 2D hard disc packing is characterized by the ratio with cubic symmetry. All embeddings are “minimally frustrated,” of the total area of the packed objects to the area of the embed- formed by removing just enough isometries of the (regular, ding space. Thus, the hexagonal “penny packing” of equal discs but unphysical) 2D hyperbolic honeycombs f3, 7g, f3, 8g, f3, 9g, 2 realizes the maximal packing density in the plane, E (24, 25). f3, 10g, and f3, 12g to allow embeddings in Euclidean 3 space. Dense packings of equal discs on the surface of the 2 sphere, Nearly all of these triangulated “simplicial polyhedra” have sym- 2, are more subtle, since the sphere’s finite area means that metrically identical vertices, and most are chiral. The most sym- S optimal solutions depend on the disc radius. The formal defini- metric examples include 10 infinite “deltahedra,” with equilateral tion of packing density for equal discs in the third homogeneous triangular faces, 6 of which were previously unknown and some 2 of which can be described as packings of Platonic deltahedra. We 2D space—the hyperbolic plane, H —is also complicated by the describe also related cubic crystalline packings of equal hyper- nature of that space. -
This Is a Set of Activities Using Both Isosceles and Equilateral Triangles
This is a set of activities using both Isosceles and Equilateral Triangles. 3D tasks Technical information This pack includes 25 Equilateral Triangles and 25 Isosceles Triangles. You will also need a small tube of Copydex. If you are new to making models with Copydex we suggest you start by watching www.atm.org.uk/Using-ATM-MATs Different polyhedra to make Kit A: 4 Equilateral Triangle MATs. Place one triangle on your desk and join the other three triangles to its three sides, symmetrically. Now add glue to one edge of each of the three added triangles so that all can be joined into a polyhedron This is your first polyhedron. Kit B: 3 Isosceles triangles and 1 equilateral triangle Kit C: 4 Isosceles triangles Kit D: 6 Equilateral Triangles Kit E: 6 Isosceles Triangles Kit F: 3 Equilateral and 3 Isosceles Triangles. (are there different possible polyhedra this time?) Kit G: 8 Equilateral Triangles. One possibility is a Regular Octahedron, but is there more than one Octahedron? Kit H: 4 Equilateral and 4 Isosceles Triangles (are there different possible polyhedra this time?) Kit I: 2 Equilateral Triangles and 6 Isosceles Triangles Kit J: Go back to the Regular Octahedron (from Kit G) and add a Regular Tetrahedron to one of its faces, then add another … until four have been added. Keep track (in a table) of how many faces and vertices the Regular Octahedron and the next four polyhedra have. Hint: if you spaced the added Tetrahedra equally around the Octahedron you will find a bigger version of a previously made solid. -
Cannonballs and Honeycombs, Volume 47, Number 4
fea-hales.qxp 2/11/00 11:35 AM Page 440 Cannonballs and Honeycombs Thomas C. Hales hen Hilbert intro- market. “We need you down here right duced his famous list of away. We can stack the oranges, but 23 problems, he said we’re having trouble with the arti- a test of the perfec- chokes.” Wtion of a mathe- To me as a discrete geometer Figure 1. An matical problem is whether it there is a serious question be- optimal can be explained to the first hind the flippancy. Why is arrangement person in the street. Even the gulf so large between of equal balls after a full century, intuition and proof? is the face- Hilbert’s problems have Geometry taunts and de- centered never been thoroughly fies us. For example, what cubic tested. Who has ever chatted with about stacking tin cans? Can packing. a telemarketer about the Riemann hy- anyone doubt that parallel rows pothesis or discussed general reciprocity of upright cans give the best arrange- laws with the family physician? ment? Could some disordered heap of cans Last year a journalist from Plymouth, New waste less space? We say certainly not, but the Zealand, decided to put Hilbert’s 18th problem to proof escapes us. What is the shape of the cluster the test and took it to the street. Part of that prob- of three, four, or five soap bubbles of equal vol- lem can be phrased: Is there a better stacking of ume that minimizes total surface area? We blow oranges than the pyramids found at the fruit stand? bubbles and soon discover the answer but cannot In pyramids the oranges fill just over 74% of space prove it. -
Ornamental Design Is… a Kind of Practical Science’ Theories of Ornament at the London School of Design and Department of Science and Art1
‘Ornamental design is… a kind of practical science’ Theories of ornament at the London School of Design and Department of Science and Art1 Elena Chestnova ‘When Semper came to England he fell in with the wrong people’ – wrote Joseph Rykwert in his commentary to Semper’s first public lecture delivered at the Metropolitan School of Design in Marlborough House.2 The ‘wrong people’ implied those associated with the Department of Science and Art: 3 Henry Cole, Richard Redgrave, the closely connected Owen Jones and Matthew Digby Wyatt, William Dyce, Ralph Wornum and other instructors linked with the controversial Design Reform of the early 1850s. Semper had been employed at the school between 1852 and 1855, and while his time in London generally is considered to have been pivotal for his theoretical work, the impact of his associations at the school has been appraised negatively and the importance of his intellectual links with the Cole circle has been downplayed.4 This approach, however, ignores a host of significant issues that formed the context for the development of Semper’s theory of design as well as 1 This article is an outcome of the workshop ‘Gottfried Semper in London 1850–55’ held at the Accademia di architettura, Università della Svizzera italiana (USI), Mendrisio, on February 26, 2014, as part of the research and edition project ‘Architecture and the Globalization of Knowledge in the 19th Century: Gottfried Semper and the Discipline of Architectural History’ headed by Sonja Hildebrand (USI, responsible) and Philip Ursprung (ETH Zurich, co-responsible) with the support of the Swiss National Science Foundation (SNSF). -
Promenade Among Words and Things: the Gallery As Catalogue, the Catalogue As Gallery
$UFKLWHFWXUDO Lending, M 2015 Promenade Among Words and Things: The Gallery as Catalogue, the Catalogue as Gallery. Architectural Histories, 3(1): 20, +LVWRULHV pp. 1–22, DOI: http://dx.doi.org/10.5334/ah.da RESEARCH ARTICLE Promenade Among Words and Things: The Gallery as Catalogue, the Catalogue as Gallery Mari Lending* In the mid-19th century new casting techniques allowed for the production of huge building fragments. Well-selected cast collections would ideally display perfect series in galleries in which the visitor could wander among monuments and experience the history of architecture on a full scale and in three dimen- sions. The disembodied material of plaster proved capable of embodying a number of modern historical taxonomies and aesthetical programs, most importantly chronology, comparison, style, and evolution. Veritable showcases of historicism, the casts could illustrate in spatial arrangements new conceptions on the history, contemporaneity and future of architecture. The plaster cast became a main medium in which to publish antiquities as novelties for grand audiences, taking the printed and the published beyond the two-dimensional space of words and images. However, due to the increasing market of casts and their sheer size and weight, the reproductions as mounted in the galleries often behaved as disorderly as architecture does outside curatorial control. In the end only the catalogues, the paper versions of these imaginary museums, could create the order their plaster referents constantly aspired to destroy. An important chapter in the history of the architecture museum, these plaster monuments belong to a part of architectural print culture in which catalogues were curated and galleries edited. -
08 … Let Us Take an Excursion Around the World!1 Monument
RA 21 239 Images 08 01. Aerial view of te Rijksmu- 09. A The Amsterdam Mara- seum & he Musempli thon crossing the passageway of Rijksmuseum 2016. … Let us take an excursion 02. Picture of the plan by J. G. B A view from the passageway 1 van Niftrik dating 1866 towards the great hall. around the world! C A view from the hall of the 03. A. Picture of the passa- Rijksmuseum and at the back of Monument and copy as geway by Cuypers showing the the image, the passageway. urban nature of it. 1885. curatorial practice in the B. Interior of passageway with its ceramic finishing, decora- international exhibitions ted with carpets, curtains and flowers, photography dating and their museums of 1935. collections, 1854-1929* 04. Drawing carried out by Pie- rre J.H. Cuypers of the northern Carolina B. García-Estévez façade, cross-section showing the passageway, the covered, In December 2018, the Victoria & Albert Museum reopened Cast illuminated courtyard and the Court after a period of intensive rehabilitation. Its collection of entrances to the museum of the plaster casts of the principal monuments and works of art is one northern façade covered by a of the very few that is still held in the original place for which they canopy. It includes the image of were designed. The impact of this encyclopaedic, exhaustive and the tramway passing through universal collection on visitors to what was then London’s South the passageway, 1885. Kensington Museum was summarized as “an excursion around the world”, in a spirit close to that which lit up the Great Exhibition of 05. -
Hyperbolic Islamic Patterns - a Beginning
BRIDGES Mathematical Connections in Art, Music, and Science Hyperbolic Islamic Patterns - A Beginning Douglas Dunham Department of Computer Science University of Minnesota,. Duluth Duluth, MN 55812-2496, USA E-mail: ddunham@d. umn . edu WebSite: http://www.d.umn.edu/-ddunham/ Abstract For more than a millennium, Islamic artists and craftsmen have used geometric patterns to decorate buildings, cloth, pottery, and other artifacts. Many of these patterns were "wallpaper" patterns - they were planar patterns that repeated in two different directions. Recently related patterns have also been drawn on the Platonic solids, which can conceptually be projected outward onto their circumscribing spheres, thus utilizing a second of the three "classical geometries". We extend this process by exhibiting repeating Islamic patterns in hyperbolic geometry, the third classical geometry. Introduction Islamic artists have long had a fascination for geometric patterns such as the one below in Figure 1 from the Alhambra palace. The purpose of this paper is to show that it is possible to create Islamic patterns in the hyperbolic plane, such as the one shown in Figure 2 which is related to the pattern of Figure 1. Figure 1: An Islamic pattern from the Alhambra. Figure 2: An Islamic hyperbolic pattern based on the Euclidean pattern of Figure 1. 248 Douglas Dunha ... The techniques for creating the original Islamic patterns were passed down from master to apprentice artisans, and have subsequently been lost. However, for more than 100 years, when it first became possible to print color reproductions, people have tried to analyze those patterns starting with Bourgoin [3]. The analysis of wallpaper patterns, patterns of the Euclidean plane that repeat in two different directions, became more precise when their 17 possible symmetry groups were classified.