非欧几何是怎么挑战“平行公设”的？
How did non-Euclidean geometry challenge the parallel postulate?
欧几里得几何假定过直线外一点有且仅有一条平行线。19 世纪数学家发现，若改成“没有”或“不止一条”平行线，仍能构造出逻辑自洽的新几何（双曲与椭圆几何）。这证明平行公设独立于其他公理，并颠覆了“几何唯一正确”的观念，为相对论时空提供了数学框架。
Euclidean geometry assumes exactly one parallel line through a point off a given line. In the 19th century mathematicians found that allowing none or more than one parallel still yields consistent new geometries (hyperbolic and elliptic). This proved the parallel postulate independent of the others and overturned the idea of one correct geometry, later providing the framework for relativity's spacetime.
难度 专业 · 适合 高中(15-18)
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#非欧几何 #平行公设 #双曲几何
射影几何中的“对偶性”是什么？
What is duality in projective geometry?
对偶性指射影几何中“点”和“直线”这两个概念可以互换而不改变定理的真假，例如“两点确定一条直线”的对偶是“两直线确定一个交点”。这种对称让许多命题成对出现，极大简化了理论的表述。对偶思想后来也影响了代数和拓扑。
Duality in projective geometry means the concepts of point and line can be swapped without changing whether a theorem is true; for example, two points determine a line dualizes to two lines determine a point. This symmetry makes many statements come in pairs and greatly simplifies the theory. The idea of duality later influenced algebra and topology.
难度 专业 · 适合 高中(15-18)
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#对偶性 #射影几何 #对称
为什么三维晶体只有 230 种空间群？
Why are there only 230 space groups in three dimensions?
晶体必须具有平移对称性，平移要构成三维点阵。点阵中允许的旋转轴被“晶体学限制定理”限制为只有 1、2、3、4、6 次（五重旋转会破坏平移结构）。把点阵对称与分子自身的对称组合并分类，恰好得到 230 个空间群，由费多罗夫和申弗利斯于 19 世纪末独立完成，是晶体学的基本语言。2011 年诺贝尔化学奖颁给准晶的发现，恰以“五重对称例外”反证了这一分类的意义。
Crystals must have translational symmetry, with translations forming a 3D lattice. The crystallographic restriction theorem limits allowed rotations to axes of order 1, 2, 3, 4, or 6 (fivefold rotation would break the lattice). Combining lattice symmetries with molecular symmetries and classifying them yields exactly 230 space groups, derived independently by Fedorov and Schönflies in the 1890s—the basic language of crystallography. The 2011 Chemistry Nobel for quasicrystals ironically highlighted fivefold-symmetric exceptions that prove the classification's significance.
难度 专业 · 适合 高中(15-18)
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#空间群 #晶体对称 #晶体学限制定理
什么是手性（左右不对称）？
What is chirality (left-right asymmetry)?
手性指一个物体与其镜像不能通过旋转和平移完全重合，就像左手和右手。具有手性的图形没有反射对称，但可能有旋转对称。手性在分子生物学中非常重要，例如左右旋的分子性质可能截然不同。
Chirality means an object and its mirror image cannot be made to coincide by rotation and translation alone, like the left and right hands. A chiral figure lacks reflection symmetry but may still have rotational symmetry. Chirality matters greatly in molecular biology, where left- and right-handed molecules can behave very differently.
难度 专业 · 适合 初中(12-15)
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#手性 #镜像 #对称
分形里的自相似对称是什么意思？
What does self-similar symmetry in fractals mean?
分形具有自相似性：无论把局部放大多少倍，它的结构和整体看起来都一样，形成一种“尺度上的对称”。雪花曲线、曼德博集合都是典型例子。这种对称不是有限次的反射或旋转，而是在无限缩小和放大中反复出现。
A fractal has self-similarity: no matter how much you zoom into a part, its structure looks like the whole, a kind of symmetry across scales. The Koch snowflake and the Mandelbrot set are classic examples. This symmetry is not a finite reflection or rotation but repeats under infinite zooming.
难度 专业 · 适合 高中(15-18)
CC BY 4.0 · 已核验 · Verified 2026-10-04 · 2 条独立证据
来源 · Source
B. B. Mandelbrot《The Fractal Geometry of Nature》, Wiley, 1982
B. B. Mandelbrot, The Fractal Geometry of Nature, Wiley, 1982
独立核验依据 · Independent verification: 核验依据 1 ISBN 9780716711865 (1982) · 核验依据 2 ISBN 9780716711865 (2021)
#分形 #自相似 #尺度对称
晶体学限制定理为什么规定：平面上的有限旋转对称阶数只能是 1、2、3、4、6？
Why does the crystallographic restriction theorem force the order of any finite rotational symmetry in the plane to be only 1, 2, 3, 4, or 6?
若平面有一个 n 重旋转对称，则把一个格点阵绕中心转 2π/n 后须仍与自身重合；代数上这要求 2cos(2π/n) 是整数，只当 n = 1,2,3,4,6 时成立（对应余弦值 2,1,0,−1,−2）。所以五重或七重旋转对称无法铺满整个平面，这也解释了自然界没有五重对称晶体。
If the plane has an n-fold rotational symmetry, rotating a lattice by 2π/n must map it onto itself; algebraically this needs 2cos(2π/n) to be an integer, which happens only for n = 1,2,3,4,6 (cosines giving 2,1,0,−1,−2). Thus five- or seven-fold rotational symmetry cannot tile the whole plane, explaining why no natural crystal has five-fold symmetry.
难度 专业 · 适合 高中(15-18)
CC BY 4.0 · 来源待核验 · Source pending review
来源 · Source
出处备注：著录含书名，但缺版次 / 出版社 / 年份中的一部分，需编辑补全后才能做独立比对。
Weyl, H., 《对称》, Princeton University Press
Weyl, H., 'Symmetry', Princeton University Press
#晶体学限制 #旋转对称 #格点 #整数约束
几何中的“对称群”是什么？
What is a symmetry group in geometry?
对称群是某个图形所有对称变换（旋转、反射等）组成的集合，这些变换之间可以组合，并满足群的公理。研究对称群能把“形状的美感”转化为严格的代数对象。例如正 n 边形的对称群有 2n 个元素，包含 n 个旋转和 n 个反射。
A symmetry group is the set of all symmetry transformations (rotations, reflections, etc.) of a figure, which can be composed and obey the axioms of a group. Studying symmetry groups turns aesthetic shape properties into rigorous algebraic objects. For example, the symmetry group of a regular n-gon has 2n elements: n rotations and n reflections.
难度 专业 · 适合 高中(15-18)
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#对称群 #群论 #几何对称
仿射变换和普通刚体变换有什么不同？
How does an affine transformation differ from rigid motion?
刚体变换（平移、旋转、反射）保持距离和角度不变，而仿射变换在保持“直线仍是直线、平行仍平行”的同时，允许拉伸、压缩和错切，从而改变长度和角度。仿射变换可表示为线性变换加平移。它在计算机图形和坐标变换中非常基础。
Rigid motions (translation, rotation, reflection) preserve distances and angles, while an affine transformation keeps lines stay lines and parallels stay parallel but allows stretching, compressing, and shearing, changing lengths and angles. An affine map is a linear transformation plus a translation. It is fundamental in computer graphics and coordinate changes.
难度 专业 · 适合 高中(15-18)
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#仿射变换 #线性变换 #错切
向量是什么，为什么有用？
What is a vector and why is it useful?
向量是同时带有大小和方向的量，如力、速度和位移，常用带箭头的线段表示。向量可以相加、数乘，并能描述平移和许多物理规律。它是物理学、计算机图形和机器学习中表示“多重信息”的基本工具。
A vector is a quantity with both magnitude and direction, such as force, velocity, and displacement, often drawn as an arrow. Vectors can be added and scaled, and they describe translations and many physical laws. They are the basic tool in physics, computer graphics, and machine learning for representing multiple pieces of information.
难度 进阶 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#向量 #方向 #线性代数
什么是相似变换？
What is a similarity transformation?
相似变换由等距变换（平移、旋转、反射）再叠加一个均匀缩放组成，保持图形形状不变而只改变大小。变换前后的图形称为相似形，对应角相等、对应边成比例。它是“缩放变换”的更一般说法，包含位置和方向调整。
A similarity transformation is an isometry (translation, rotation, reflection) followed by a uniform scaling, preserving shape while changing only size. The resulting figures are similar, with equal angles and proportional sides. It is the more general term for scaling that also allows position and orientation changes.
难度 进阶 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#相似变换 #缩放 #相似形
极坐标系和普通直角坐标系有什么不同？
How does the polar coordinate system differ from the Cartesian one?
直角坐标系用（x, y）表示点到两条垂直轴的距离，而极坐标用（r, θ）表示点到原点的距离和方向与水平轴的夹角。极坐标特别适合描述圆形、螺旋和旋转对称图形。同一曲线在不同坐标系下方程形式可能大不相同。
The Cartesian system uses (x, y) as distances to two perpendicular axes, while polar coordinates use (r, θ) as the distance from the origin and the angle from the horizontal axis. Polar coordinates suit circles, spirals, and rotationally symmetric figures. The same curve can look very different in equation form across systems.
难度 进阶 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#极坐标 #坐标系 #角度
反射变换（镜像）在几何中如何描述？
How is a reflection (mirror) transformation described in geometry?
反射变换把图形关于一条直线（镜面）翻转，使每一点落到另一侧等距的位置。它保持图形的形状和大小不变，但会改变“手性”（左右对调）。反射是等距变换的一种，常用于研究对称图形和晶体结构。
A reflection flips a figure across a line (the mirror), sending each point to an equidistant position on the other side. It preserves shape and size but reverses handedness (left and right swap). Reflection is an isometry and is useful for studying symmetric figures and crystal structures.
难度 进阶 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#反射 #镜像 #等距变换
为什么墙纸的花纹图案只有 17 种不同的对称类型？
Why are there only 17 different symmetry types of wallpaper patterns?
墙纸图案的对称由平移、旋转、反射、滑移反射四种等距变换的组合决定。把所有这些对称操作看成一个群，并限定平移必须构成二维格子，就能完整分类出恰好 17 个不同的“墙纸群”。同样的思想延伸三维，就得到晶体学中重要的 230 个空间群。
Wallpaper symmetry comes from combinations of four isometries: translations, rotations, reflections, and glide reflections. Treating all symmetry operations as one group, with translations required to form a two-dimensional lattice, yields exactly 17 distinct “wallpaper groups”. The same idea in three dimensions produces the 230 space groups central to crystallography.
难度 进阶 · 适合 初中(12-15)
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#墙纸群 #对称群 #等距变换
平移变换是什么意思？
What does a translation transformation mean?
平移是把图形沿某个方向移动固定距离，图形的每个点都移动相同的向量。平移不改变图形的形状、大小和方向，只改变位置。它是向量加法在几何中的直观体现。
A translation slides a figure by a fixed distance in a direction, moving every point by the same vector. It changes only the position, not the shape, size, or orientation. It is the geometric picture of vector addition.
难度 进阶 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#平移 #向量 #等距变换
什么是平移对称？
What is translational symmetry?
平移对称指图形沿某个方向平移一定距离后与原图重合，就像无限延伸的条纹或瓷砖图案。把图形不断平铺就会得到周期性的重复结构。平移对称是晶体和装饰图案中最常见的对称类型之一。
Translational symmetry means a figure coincides with itself after sliding a certain distance in a direction, like an infinitely repeating stripe or tiled pattern. Repeating the slide yields a periodic structure. It is among the most common symmetries in crystals and decorative patterns.
难度 进阶 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#平移对称 #周期性 #平铺
旋转变换有哪些关键要素？
What are the key elements of a rotation transformation?
旋转变换需要三个要素：旋转中心、旋转角度和旋转方向（顺时针或逆时针）。图形绕中心转动后，形状和大小不变，只是位置改变。旋转对称图形能在转动某个角度后仍与原图重合，例如正方形转 90 度不变。
A rotation needs three elements: a center, an angle, and a direction (clockwise or counterclockwise). After turning around the center, the figure keeps its shape and size but changes position. A rotationally symmetric figure coincides with itself after some angle, such as a square unchanged by 90 degrees.
难度 进阶 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#旋转 #旋转中心 #角度
等边三角形有哪些对称操作？它们一共有多少个？
What symmetry operations does an equilateral triangle have, and how many are there in total?
等边三角形有 6 个对称操作：绕中心旋转 0°、120°、240° 三种，加上沿三条对称轴（每条轴过顶点和对边中点）的镜像翻折三种。把任意两个操作接连执行，结果仍是这 6 个之一，它们构成一个群，叫二面体群 D3。旋转和翻折合在一起，能完全刻画三角形的对称性。
An equilateral triangle has 6 symmetries: three rotations about its center by 0°, 120°, and 240°, plus three reflections across axes through a vertex and the midpoint of the opposite side. Composing any two of these always yields another of the six, and they form a group called the dihedral group D3. Together, rotations and reflections capture the triangle's symmetry completely.
难度 进阶 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-10-03
来源 · Source
《什么是数学》(柯朗)（对称与群）
What Is Mathematics? (Courant), symmetry and groups
#对称群 #二面体群 #旋转与镜像
什么是中心对称？
What is central symmetry?
中心对称指图形绕一个中心点旋转 180 度后与原图完全重合。平行四边形、圆和正六边形都具有中心对称性。和轴对称不同，中心对称关注的是“绕点翻转”而非“沿轴折叠”。
Central symmetry means a figure coincides with itself after a 180-degree rotation about a central point. Parallelograms, circles, and regular hexagons are centrally symmetric. Unlike axial symmetry, it is about turning around a point rather than folding across a line.
难度 进阶 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#中心对称 #旋转180度 #对称
什么是自相似？
What is self-similarity?
自相似指一个整体对象的局部在放大后与整体形状相同，常见于海岸线、山脉和某些数学曲线。它意味着图形在不同尺度上重复出现相同的结构。自相似是分形和许多自然形态的重要特征。
Self-similarity means a part of an object, when enlarged, has the same shape as the whole, seen in coastlines, mountains, and certain mathematical curves. It means the same structure repeats across scales. Self-similarity is a key feature of fractals and many natural forms.
难度 进阶 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-10-04 · 2 条独立证据
来源 · Source
B. B. Mandelbrot《The Fractal Geometry of Nature》, Wiley, 1982
B. B. Mandelbrot, The Fractal Geometry of Nature, Wiley, 1982
独立核验依据 · Independent verification: 核验依据 1 ISBN 9780716711865 (1982) · 核验依据 2 ISBN 9780716711865 (2021)
#自相似 #分形 #尺度
平面上的“壁纸群”一共有多少种，为什么只有这么多？
How many wallpaper groups are there in the plane, and why exactly that many?
平面上只靠平移、旋转、反射、滑反射四种等距变换重复的图案，本质上只有 17 种对称类型，称为 17 个壁纸群。这是由二维空间等距群的代数结构决定的（晶体学限制使旋转角只能是 60°、90°、120°、180°），完整分类由费多洛夫等人于 19 世纪末完成。
Planar patterns repeated by the four isometries—translation, rotation, reflection, glide reflection—fall into exactly 17 symmetry types, the 17 wallpaper groups. This is forced by the algebra of the plane's isometry group (the crystallographic restriction limits rotation angles to 60°, 90°, 120°, 180°), and the full classification was completed by Fedorov and others in the late 19th century.
难度 进阶 · 适合 高中(15-18)
CC BY 4.0 · 已核验 · Verified 2026-11-02
来源 · Source
Grünbaum, B. 与 Shephard, G. C., 《镶嵌与图案》, W. H. Freeman
Grünbaum, B. & Shephard, G. C., 'Tilings and Patterns', W. H. Freeman
独立核验依据 · Independent verification: 核验依据 1 OPENLIBRARY 标识 /works/OL8056246W：Tilings and patterns（首版
#壁纸群 #晶体学限制 #平面对称 #17种
什么是对称？
What is symmetry?
对称指一个图形或物体经过某种变换（如翻转、旋转）后，看起来和原来一样。常见的对称包括左右镜像对称和绕中心旋转对称。对称的图形往往让人感到和谐美观，也常见于自然界和艺术中。
Symmetry means a shape or object looks the same after some transformation, such as flipping or rotating. Common types include left-right mirror symmetry and rotation about a center. Symmetric forms often feel harmonious and appear frequently in nature and art.
难度 入门 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#对称 #变换 #几何
什么是全等图形？
What are congruent figures?
全等图形指形状和大小完全相同、能够完全重合的两个图形，可以通过平移、旋转、反射等刚体变换相互得到。全等不关心位置，只关心几何本质一致。判定三角形全等有边角边的若干标准方法。
Congruent figures are two shapes that are exactly the same in form and size and can coincide completely, obtainable from each other by rigid motions like translation, rotation, and reflection. Congruence ignores position and cares only about identical geometry. Triangles have standard congruence criteria such as side-angle-side.
难度 入门 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#全等 #刚体变换 #几何
什么是旋转对称？
What is rotational symmetry?
旋转对称指图形绕某个中心点转动一定角度后，仍和原来的样子重合。转动一周所需的最小角度能整除 360 度的图形具有旋转对称，例如风车叶片。圆形则无论转多少度都看起来一样。
Rotational symmetry means a figure coincides with itself after turning by some angle around a center. A figure whose smallest such angle divides 360 degrees has rotational symmetry, like windmill blades. A circle looks the same at any angle.
难度 入门 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#旋转对称 #旋转角 #几何
为什么蜜蜂把蜂巢做成六边形的小格子，而不是正方形或三角形？
Why do bees build honeycomb from hexagonal cells instead of squares or triangles?
六边形能像正方形、三角形一样严丝合缝地铺满平面（密铺），但在同样周长下围成的面积最大、也最省筑巢的蜡。大自然用“最省材料”的形状，于是蜂巢成了六边形——这既是数学里的平面密铺，也是动物智慧的例子。
A hexagon tiles the plane without gaps like squares or triangles, but encloses the most area for a given perimeter, saving wax. Nature picks the most material-efficient shape, so honeycomb is hexagonal—both a planar tiling in math and an example of animal ingenuity.
难度 入门 · 适合 幼儿(3-6)
CC BY 4.0 · 已核验 · Verified 2026-11-02
来源 · Source
Fejes Tóth, L., 《平面与空间中的密铺与堆积》, Springer
Fejes Tóth, L., 'Lagerungen in der Ebene, auf der Kugel und im Raum', Springer
独立核验依据 · Independent verification: 核验依据 1 OPENLIBRARY 标识 /works/OL36644774W：Lagerungen in der Eben
#蜂巢 #六边形 #密铺 #最省材料
什么是坐标系？
What is a coordinate system?
坐标系是一套用数字或数组来标定点的位置的规则，最常见的平面坐标系用两个数（横、纵坐标）确定一个点。它把几何问题转化为代数计算，使画图、导航和建模成为可能。除了直角坐标，还有极坐标、三维坐标等多种坐标系。
A coordinate system is a set of rules using numbers or arrays to locate points; the commonest planar one uses two numbers (horizontal and vertical coordinates) to fix a point. It turns geometric problems into algebra, enabling drawing, navigation, and modeling. Besides Cartesian, there are polar, 3D, and other coordinate systems.
难度 入门 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#坐标系 #位置 #几何代数