Tessellabilities, Reversibilities, and Decomposabilities of Polytopes

28/08/2013
Auteurs : Jin Akiyama
OAI : oai:www.see.asso.fr:2552:5590
DOI :

Résumé

Tessellabilities, Reversibilities, and Decomposabilities of Polytopes

Média

Voir la vidéo

Métriques

197
9
2.63 Mo
 application/pdf
bitcache://0b73b5b82af1ef5c4e34c20e8d806a16b94c6d92

Licence

Creative Commons Aucune (Tous droits réservés)

Sponsors

Sponsors scientifique

logo_smf_cmjn.gif

Sponsors financier

logo_gdr-mia.png
logo_inria.png
image010.png
logothales.jpg

Sponsors logistique

logo-minesparistech.jpg
logo-universite-paris-sud.jpg
logo_supelec.png
Séminaire Léon Brillouin Logo
logo_cnrs_2.jpg
logo_ircam.png
logo_imb.png
<resource  xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance"
                xmlns="http://datacite.org/schema/kernel-4"
                xsi:schemaLocation="http://datacite.org/schema/kernel-4 http://schema.datacite.org/meta/kernel-4/metadata.xsd">
        <identifier identifierType="DOI">10.23723/2552/5590</identifier><creators><creator><creatorName>Jin Akiyama</creatorName></creator></creators><titles>
            <title>Tessellabilities, Reversibilities, and Decomposabilities of Polytopes</title></titles>
        <publisher>SEE</publisher>
        <publicationYear>2014</publicationYear>
        <resourceType resourceTypeGeneral="Text">Text</resourceType><dates>
	    <date dateType="Created">Fri 3 Jan 2014</date>
	    <date dateType="Updated">Sun 25 Dec 2016</date>
            <date dateType="Submitted">Fri 25 May 2018</date>
	</dates>
        <alternateIdentifiers>
	    <alternateIdentifier alternateIdentifierType="bitstream">0b73b5b82af1ef5c4e34c20e8d806a16b94c6d92</alternateIdentifier>
	</alternateIdentifiers>
        <formats>
	    <format>application/pdf</format>
	</formats>
	<version>25064</version>
        <descriptions>
            <description descriptionType="Abstract"></description>
        </descriptions>
    </resource>
.

Tessellabilities, Reversibilities, and Decomposabilities of Polytopes ― A Survey ― École nationale supérieure des mines de Paris Paris, August 28, 2013 Jin Akiyama: Tokyo University of Science Ikuro Sato: Miyagi Cancer Center Hyunwoo Seong: The University of Tokyo Tokyo, Japan 1. P1-TILES AND P2-TILES 2 3 A P1-tile is a polygon which tiles the plane with translations only. Two families of convex P1-tiles : (1) parallelograms and (2) hexagons with three pairs of opposite sides parallel and of the same lengths (P1-hexagons). Parallelogram P1-hexagon Parallelepiped(PP) Rhombic Dodecahedron(RD) Hexagonal Prism(HP) Elongated Rhombic Dodecahedron(ERD) Truncated Octahedron(TO) 4 F1 F2 F3 F4 F5 A 3-dimensional P1-tile is a polyhedron which tiles the space with translations only. Five families of convex 3-dimensional P1-tiles (Fedorov) : 10 Triangle Quadrilateral P2-pentagon (BC∥ED) P2-hexagon (QPH) (AB∥ED and |AB|=|ED|) Theorem A Every convex P2-tile belongs to one of the following four families: F1 F2 F3 F4 A P2-tile is a polygon which tiles the plane by translations and 180° rotations only. 11 Determine all convex 3-dimensional P2-tiles, i.e., convex polyhedra each of which tiles the space in P2-manner. (cf) triangular prism, … A net of a convex polyhedron P is defined to be a connected planar object obtained by cutting the surface of P. An ART (almost regular tetrahedron) is a tetrahedron with four congruent faces. CG Theorem B (J.A(2007)) Every net (convex or concave) of an ART tiles the plane in P2-manner. Artworks Artworks Artworks 2. REVERSIBILITY 17 18 Volvox, a kind of green alga known as one of the most simple colonial (≒ multicellular) organisms, reproduces itself by reversing its interior offspring and its surface. Theorem C (J.A. (2007)) If a pair of polygons A and B is reversible, then each of them tiles the plane by translations and 180°rotations only (P2- tiling). 19 A : red quadrilateral, B: blue triangle CG CG 20 Let Π be the set of the five Platonic 1, σ2, σ3, σ4. Then Φ = {σ1, . . ., σ4} is an element set for Π, and the decomposition of each Platonic solid into these elements is summarized in Table 3. Theorem D ( J.A., I. Sato, H. Seong (2013)) For an arbitrary convex P2-tile P and an arbitrary family Fi (i= 1, 2, 3, and 4) of convex P2-tiles, there exists a polygon Q ∈ Fi such that the pair P and Q is reversible. A king in a cage 22 Spider ⇔ Geisha 23 A 3-dimensional P1-tile is said to be canonical if it is convex and symmetric with respect to each orthogonal axis. F5F4F3F2F1 24 UFO ⇔ Alien CG Let Π be the set of the five Platonic 1, σ2, σ3, σ4. Then Φ = {σ1, . . ., σ4} is an element set for Π, and the decomposition of each Platonic solid into these elements is summarized in Table 3. Theorem E ( J.A., I. Sato, H. Seong (2011)) For an arbitrary canonical 3-dimensional P1-tile P and an arbitrary family Fi (i= 1, 2, 3, 4, and 5) of canonical 3- dimensional P1-tiles, there exists a polyhedron Q ∈ Fi such that the pair P and Q is reversible. Cube -> Hexagonal Prism 25 CG Hexagonal Prism -> Truncated Octahedron Rhombic Dodecahedron -> Elongated Rhombic Dodecahedron CG CG 3. TILINGS AND ATOMS 26 2 2 2 6 6 2 6 2 2 2 4 4 3 2 23 A symmetric pair of pentadra Pentadron is a convex pentahedron whose net is as follows: 28 Tetrapak is a special kind of ART(tetrahedron with four congruent faces) made by pentadra as follows: 29 Theorem F (J.A.) A tetrapak tiles the space and its net tiles the plane. Problem Determine all convex polyhedra, each of which tiles the space and one of its nets tiles the plane. Theorem G (J.A, G.Nakamura, I.Sato (2012)) Every convex 3-dimensional P1-tile (or its affine- stretching transform) can be constructed by copies of a pentadron. 31 Cube Hexagonal prism 32 33 Truncated octahedron Rhombic dodecahedron 35 Elongated rhombic dodecahedron