多面体の非同型な展開図の個数について  [in Japanese] The Number of Different Unfoldings of Polyhedra  [in Japanese]

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Abstract

多面体の展開図(辺展開とも呼ばれる)は,多面体を辺に沿って切り開くことで得られる多角形である.切り開く辺が異なっても,同型な展開図が得られることがある.例えば,立方体には384通りの展開の仕方(つまり辺の切り開き方)があるが,同型なものを除去することで,11種類の本質的に異なる(非同型な)展開図が得られる.本稿では,任意の多面体に対し,非同型な展開図の個数を数え上げる方法について述べる.また,この手法をすべての整面凸多面体(正多面体,半正多面体,ジョンソン・ザルガラーの多面体,アルキメデスの角柱と反角柱)に適用し,それぞれの非同型な展開図の個数を示す.たとえば,角切り二十面体(サッカーボールフラーレン)には375,291,866,372,898,816,000通りの展開方法があるが,同型なものを排除することで3,127,432,220,939,473,920種類の異なる展開図が存在することが分かった.

An unfolding (also called an edge unfolding) of a polyhedron is a simple polygon obtained by cutting along the edges of the polyhedron and unfolding it into a plane. Different edge-cuts of a polyhedron may have the same (i.e., isomorphic) unfolding. For example, a cube has 384 way of unfolding (i.e., cutting its edges). By omitting mutually isomorphic unfoldings, we have 11 essentially different (i.e., nonisomorphic) unfoldings. In this paper, we propose how to count the number of nonisomorphic unfoldings for any polyhedron. We also give the number of nonisomorphic unfoldings for all regular-faced convex polyhedra (i.e., Platonic solids, Archimedean solids, Johnson-Zalgaller solids, Archimedean prisms, and antiprisms). For examaple, while a truncated icosahedron (a Buckminsterfullerene. or a soccer ball fullerene) has 375,291,866,372,898,816,000 way of unfolding, it has 3,127,432,220,939,473,920 nonisomorphic unfoldings. (This article is a technical report without peer review.)

Journal

  • IEICE technical report. Theoretical foundations of Computing

    IEICE technical report. Theoretical foundations of Computing 112(24), 59-66, 2012-05-07

    The Institute of Electronics, Information and Communication Engineers

References:  17

Codes

  • NII Article ID (NAID)
    110009569455
  • NII NACSIS-CAT ID (NCID)
    AN10013152
  • Text Lang
    JPN
  • Article Type
    ART
  • ISSN
    0913-5685
  • NDL Article ID
    023742646
  • NDL Call No.
    Z16-940
  • Data Source
    CJP  NDL  NII-ELS 
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