内容概述
故事列举了关于正多面体的 8 个思考题：(1) 正多面体定义中的三个性质——正多边形面、全等的面、全等的多面角——各自必需；(2) 由这三条可推出多面角正则，因此定义可简化为两条；(3) 内接同一球的正十二面体与正二十面体，反而是正十二面体体积更大，立方体也比正八面体体积大；(4) 内接同一球的正十二面体与正二十面体有共同的内接球，立方体与正八面体亦然；(5) 对给定表面积，正八面体体积是否最小、正二十面体是否最大；(6) 内接同一球的正十二面体与正二十面体体积之比，等于同内接球的立方体边长与正二十面体边长之比；(7) 海普希克勒斯指出，内接同一球的正十二面体与正二十面体体积之比等于表面积之比；(8) 它们各面的外接圆相等。
中文正文
65° 柏拉图体的问题
（1）前面关于正多面体的定义，包含 3 个性质：正多边形的面，全等的面，全等的多面角。许多立体几何课本没有列出全部性质。感兴趣的读者也许可以通过几个反例来证明那 3 个条件都是必需的。
（2）根据（1）列举的 3 个性质，我们可以发现多面角都是正的。请读者自己证明这一点，这也就证明 3 个性质可以归结为 2 个：正多边形的面和正多面角。
（3）没经验的人几乎直觉地相信，内接于同一个球的正 12 面体（有 12 个面）与正 20 面体（有 20 个面），正 20 面体的体积更大。请读者证明，事实恰好相反；读者还可以证明，内接于同一个球的立方体（6 个面）和正 8 面体，立方体的体积更大。
（4）有趣的是，内接于同一个球的正 12 面体和正 20 面体有着共同的内接球。请读者证明这一点，并说明它对于内接于同一个球的立方体和正 8 面体也是正确的。
（5）前面说过，开普勒凭直觉假定，对一定的表面积，正 8 面体包容了最小的体积，正 20 面体包容了最大的体积，真是这样吗？
（6）如果正 12 面体、正 20 面体和立方体内接于同一个球，请证明正 12 面体的体积与正 20 面体的体积之比，等于立方体边长与正 20 面体的边长之比。
（7）欧几里得《原本》第 14 卷指出，海普希克勒斯（Hypsicles）发现，如果正 12 面体与正 20 面体内接于同一个球，那么二者的体积之比等于表面积之比。请证明。
（8）证明内接于同一个球的正 12 面体和正 20 面体具有相等的表面的周长。
英文正文
65° Some problems concerning the Platonic solids.
(1) In Item 61°, the definition of regularity of a polyhedron involves three properties: regular faces, congruent faces, congruent polyhedral angles Many textbooks on solid geometry do not give all three of the defining properties. The interested reader may care to show, by counterexamples, that all three properties are necessary.
(2) From the three defining properties listed in (1), one can deduce the regularity of the polyhedral angles. The reader is invited to do this, and then to show that the three defining properties can be replaced by only two: regular faces and regular polyhedral angles.
(3) The uninitiated will almost always intuitively believe that of the regular dodecahedron (a solid having 12 faces) and a regular icosahedron (a solid having 20 faces) inscribed in the same sphere, the icosahedron has the greater volume. The reader is invited to show that the reverse is actually the case, and also to show that of a cube (a solid having 6 faces) and a regular octahedron (a solid having 8 faces) inscribed in the same sphere, the cube has the larger volume.
(4) It is interesting that a regular dodecahedron and a regular icosahedron inscribed in the same sphere have a common inscribed sphere. Prove this, and show that the same is true of a cube and a regular octahedron inscribed in the same sphere.
(5) In Item 62° we noted that Kepler intuitively assumed that, of the five regular solids, for a given surface area the tetrahedron encloses the smallest volume and the icosahedron encloses the largest volume. Is this so?
(6) A regular dodecahedron, a regular icosahedron, and a cube are inscribed in the same sphere. Prove that the volume of the dodecahedron is to the volume of the icosahedron as the length of an edge of the cube is to the length of an edge of the icosahedron.
(7) In the so-called Book XIV of Euclid's Elements, it is pointed out that Hypsicles observed that if a regular dodecahedron and a regular icosahedron are inscribed in the same sphere, then their volumes are in the same ratio as their surface areas. Prove this.
(8) Show that the circumcircles of the faces of the regular dodecahedron and the regular icosahedron inscribed in the same sphere are equal.
英语学习拓展
常用词汇
polyhedron音标/pɒliˈhiːdrən/
多面体。
原文用法：the definition of regularity of a polyhedron
regularity音标/ˌreɡjʊˈlærəti/
正则性；规则性。
原文用法：the definition of regularity of a polyhedron
congruent音标/ˈkɒŋɡruənt/
全等的。
原文用法：congruent faces, congruent polyhedral angles
polyhedral音标/ˌpɒliˈhiːdrəl/
多面体的。
原文用法：congruent polyhedral angles
properties音标/ˈprɒpətiz/
性质（复数）。
原文用法：three properties
counterexamples音标/ˌkaʊntərɪɡˈzɑːmplz/
反例（复数）。
原文用法：by counterexamples, that all three properties are necessary
deduce音标/dɪˈdjuːs/
推导。
原文用法：one can deduce the regularity of the polyhedral angles
inscribed音标/ɪnˈskraɪbd/
内接的。
原文用法：inscribed in the same sphere
circumcircles音标/ˈsɜːkəmˌsɜːklz/
外接圆（复数）。
原文用法：the circumcircles of the faces
同义词与近义词
对应核心词汇：properties
对应核心词汇：properties。
辨析：properties 指数学对象的内在性质，characteristics 更强调区分性特征。
对应核心词汇：deduce
对应核心词汇：deduce。
辨析：deduce 强调从一般前提出发严密推出结论，infer 更宽泛。
常用短语与固定搭配
- be inscribed in：内接于。原文用法：inscribed in the same sphere
- the same ... as ...：与……相同的……。原文用法：their volumes are in the same ratio as their surface areas
- be replaced by：被……替代。原文用法：the three defining properties can be replaced by only two
- in the same ratio as：与……成相同比例。原文用法：their volumes are in the same ratio as their surface areas
常用句式
- The reader is invited to ...：请读者……。原文例句：The reader is invited to show that the reverse is actually the case.
- Prove that ...：证明……。原文例句：Prove that the volume of the dodecahedron is to the volume of the icosahedron as the length of an edge of the cube is to the length of an edge of the icosahedron.
长难句解析
The uninitiated will almost always intuitively believe that of the regular dodecahedron (a solid having 12 faces) and a regular icosahedron (a solid having 20 faces) inscribed in the same sphere, the icosahedron has the greater volume.
- 主干：The uninitiated will almost always intuitively believe that ... the icosahedron has the greater volume。
- 修饰与从句：that 引导宾语从句；of the regular dodecahedron and a regular icosahedron 表示比较范围；两个 a solid having ... faces 为同位语；inscribed in the same sphere 为过去分词短语，修饰 dodecahedron 和 icosahedron。
- 逻辑关系：指出外行者常会凭直觉认为内接同一球的正二十面体体积更大，为后文“事实相反”埋下伏笔。
语言使用特点
- 问题描述大量使用祈使句和被动结构（Prove that ..., Show that ...），体现数学证明题的客观性与指令性。
迁移练习
- 用英文写出正多面体定义中的三个性质。
- 用 The reader is invited to ... 改写一句数学问题提示。
点击展开参考答案
- regular faces, congruent faces, congruent polyhedral angles.
- 示例：The reader is invited to prove that the sum of the interior angles of a triangle equals two right angles.