An Infinite Class of Deltahedra
An Infinite Class of Deltahedra
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无限类 Deltahedra
DOI：
10.1080/0025570x.1978.11976675
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发表时间：
1978
影响因子：
--
通讯作者：
C. W. Trigg
中科院分区：
文献类型：
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作者：
C. W. Trigg
Cundy [1]提出了一个合适的名称，即三角面体（A-hedra），用于所有面都是等边三角形的多面体。三个正（柏拉图）的固体，四面体，八面体和二十面体是凸三角形，分别有4，8和20个等边三角形面。三角形的模型可以很方便地通过使用10号橡皮筋连接3英寸法兰等边三角形纸板面板发明的西雅图建筑师，弗雷德Bassetti [2]。这些“Poly-O板”的试剂盒可以从Book-Lab，Inc. [3]的文件。或者，自制的面板可以从铁路板切割的方式清楚地描述了斯图尔特[4]，普里切特[5]和伍拉弗[6]。三个凸三角形可以通过将两个全等金字塔的底部放在一起而形成：6面三角形双金字塔或双四面体，8面正方形双金字塔或八面体和10面五边形双金字塔。六个等边三角形可以组合成一个正六边形。两个这样的六边形在它们的周边连接形成一个退化的双金字塔。如果沿着沿着两条重合的长对角线将其分成两部分，则可以通过向内按压对角线的端部来打开这两部分。其中一个打开的部分旋转900度后，两个部分可以重新连接，形成一个凸形的12面连体十二面体。只有八个凸三角形[7]。完整的一套是14面的三边三棱柱，通过将正方形金字塔连接到三棱柱的正方形侧面而形成;和16面的双帽正方形反棱镜，通过将正方形金字塔连接到正方形反棱镜的两个底座而形成。因此，八个凸三角形由4、6、8、10、12、14、16和20个等边三角形形成。没有18个面的凸三角形，也没有超过20个面的凸三角形。一旦放弃凸性，形成三角形的可能性是无穷无尽的。有一种特别有趣和吸引人的类型是通过将正四面体连接到其他三角体的部分或全部面上而形成的。这样通过增广而形成的三角形是伪星形的，因为它们与通过扩展基本多面体的面的平面而产生的固体不同，除了增广八面体的情况，即开普勒著名的星形八面体。一个特别有吸引力的模型是伪星状二十面体。三角形的一个无限子类，螺旋（扭曲）类型的模型，可以从等边三角形（图1）的公共边弯曲的条带构造。任何所需长度的这些条可以很容易地形成纸板面板橡胶带的方法。螺旋三角形的模型可以通过连接三个带的最左边的角的边来形成三面角来开始。通过连接具有共同端部的相邻条带的三角形的边来进行构造。（彩色条纹强调三角面体的螺旋性质。这三个条带可以从Hand L图案中以四种方式选择，即：HHH、LLL、HHL和LLH。HHH形式看起来向右扭曲（参见图2），而LLL形式看起来向左扭曲。然而，一般来说，每一种形式都是由一堆规则的八面体组成，
The appropriate designation deltahedra (A-hedra) has been proposed by Cundy [1] for polyhedra all faces of which are equilateral triangles. Three of the regular (Platonic) solids the tetrahedron, octahedron, and icosahedron are convex deltahedra with 4, 8, and 20 equilateral triangular faces, respectively. Models of deltahedra can be made conveniently by using No. 10 rubber bands to join the 3-inch flanged equilateral triangle cardboard panels invented by Seattle architect, Fred Bassetti [2]. Kits of these "Poly-O panels" can be purchased from Book-Lab, Inc. [3]. Or, home-made panels can be cut from railroad board in the manner clearly described by Stewart [4], Pritchett [5], and Woolaver [6]. Three convex deltahedra can be formed by placing together the bases of two congruent pyramids: the 6-faced triangular dipyramid or ditetrahedron, the 8-faced square dipyramid or octahedron, and the 10-faced pentagonal dipyramid. Six equilateral triangles can be assembled into a regular hexagon. Two such hexagons joined around their perimeters form a degenerate dipyramid. If this is separated into two parts along two coincident long diagonals, the two parts can be opened up by pressing inward on the ends of the diagonals. After one of the opened parts is rotated through 900, the two parts can be rejoined to form a 12-faced Siamese dodecahedron which is convex. There are only eight convex deltahedra [7]. Completing the set are the 14-faced triaugmented triangular prism, formed by attaching square pyramids to the square lateral faces of a triangular prism; and the 16-faced dicapped square antiprism, formed by attaching square pyramids to the two bases of a square antiprism. Thus the eight convex deltahedra are formed from 4, 6, 8, 10, 12, 14, 16 and 20 equilateral triangles. There are no convex deltahedra with 18 faces, nor any with more than 20 faces. Once convexity is abandoned, the possibilities of forming deltahedra are endless. Members of one particularly interesting and attractive type are made by attaching regular tetrahedra to some or all of the faces of other deltahedra. The deltahedra thus made by augmentation are pseudo-stellated, since they are not the same as the solids produced by extending the planes of the faces of the basic polyhedron, except in the case of the augmented octahedron which is Kepler's famous stella octangula. A particularly attractive model is the pseudo-stellated icosahedron. Models of an infinite subclass of deltahedra, the spiral (twisted) type, can be constructed from strips of equilateral triangles (FIGURE 1) flexing about the common sides. Any desired lengths of these strips can be formed easily by the cardboard panel-rubber band method. Models of spiral deltahedra can be started by joining the sides of the left-most angles of three strips to form a trihedral angle. The construction proceeds by joining sides of triangles of adjacent strips that have an end in common. (Differently colored strips emphasize the spiral nature of the deltahedron.) The three strips may be chosen from the Hand L-patterns in four ways, namely: HHH, LLL, HHL, and LLH. The HHH form appears to twist to the right (see FIGURE 2), and the LLL form appears to twist to the left. However, in general, each form consists of a pile of regular octahedra capped top and bottom with a