两点之间小球最快滑下的路径是直线吗？
Is the straight line the fastest path for a ball to slide between two points?
不是。最速降线是一条摆线（圆滚线）：先陡降积累速度，再用曲线抄近路，总用时比直线更短。这是变分法的经典问题。
No. The brachistochrone is a cycloid: it drops steeply first to build speed, then curves to shorten the path, beating the straight line in time. A classic calculus-of-variations problem.
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
Morris Kline,《Mathematical Thought from Ancient to Modern Times》Oxford University Press,1972;中译《古今数学思想》上海科学技术出版社
M. Kline, Mathematical Thought from Ancient to Modern Times, Oxford University Press, 1972; Chinese ed. Shanghai Scientific & Technical Publishers
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL1947921W：Mathematical Thought fr
#最速降线 #摆线 #变分法
埃舍尔的圆极限系列与圆内的双曲几何有什么关系？
How are Escher's Circle Limit works related to hyperbolic geometry in a disk?
它们描绘了庞加莱圆盘模型：圆内越靠边距离越大，天使与魔鬼的尺寸在欧氏眼里缩小、在双曲意义下却相等。这是双曲平面在有限圆盘里的投影。
They depict the Poincaré disk model: near the rim distance grows, so angels and devils shrink in Euclidean view but are equal in hyperbolic sense. It is a hyperbolic plane projected into a finite disk.
难度 专业 · 适合 高中(15-18)
CC BY 4.0 · 已核验 · Verified 2026-10-04
来源 · Source
Greenberg《Euclidean and Non-Euclidean Geometries》；Euclid《几何原本》
Greenberg, Euclidean and Non-Euclidean Geometries; Euclid, Elements
独立核验依据 · Independent verification: 核验依据 1 ISBN 9780665590955 (2002)
#双曲几何 #庞加莱圆盘 #埃舍尔
埃舍尔的镶嵌画用了什么数学原理？
What mathematical principle do Escher's tessellations use?
他用平移、旋转、反射等对称把平面无缝铺满（密铺），常借助群论中的壁纸群描述重复模式。人与飞鸟可互为背景地填满整个平面。
He used translations, rotations, and reflections to tile the plane seamlessly (tessellation), often described by wallpaper groups in group theory. Figures and background interlock to fill the plane.
难度 进阶 · 适合 大众
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
#埃舍尔 #密铺 #对称群
彭罗斯镶嵌奇在哪？
What is strange about Penrose tiling?
它用两种菱形就能铺满平面却绝不出现周期性重复，且具有五重旋转对称。这种‘准晶体’结构后来在真实材料中得到印证。
It tiles the plane with just two rhombus shapes yet never repeats periodically, and shows fivefold rotational symmetry. Such 'quasicrystal' structure was later found in real materials.
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
#镶嵌 #准晶
等周问题说给定周长，什么图形面积最大？
The isoperimetric problem: for a given perimeter, which shape has the largest area?
答案是圆。平面上给定周长的所有闭曲线中，圆围成面积最大；反之给定面积时圆周长最短。它可用变分法或对称化严格证明。
The answer is the circle: among all closed curves with fixed perimeter, the circle encloses the maximum area, and conversely it minimizes perimeter for fixed area. Proved via calculus of variations or symmetrization.
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
#等周 #圆 #变分法
阿基米德螺旋有什么性质，和等角螺线有何不同？
What are the properties of the Archimedean spiral, and how does it differ from a logarithmic spiral?
阿基米德螺线满足极径与极角成正比（r=aθ），相邻圈间距恒定，常用于卷尺与凸轮。等角螺线r=a^θ的圈间距则按几何级数增长。
The Archimedean spiral has r=aθ, so successive turns are evenly spaced — used in roll tapes and cams. The logarithmic spiral r=a^θ spaces its turns geometrically.
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
Morris Kline,《Mathematical Thought from Ancient to Modern Times》Oxford University Press,1972;中译《古今数学思想》上海科学技术出版社
M. Kline, Mathematical Thought from Ancient to Modern Times, Oxford University Press, 1972; Chinese ed. Shanghai Scientific & Technical Publishers
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL1947921W：Mathematical Thought fr
#阿基米德 #螺旋 #极坐标
蜜蜂蜂巢为什么是六边形而不是正方形或三角形？
Why are honeycombs hexagonal rather than square or triangular?
在给定周长下，六边形能以最少蜡壁围出最大面积，且能无空隙铺满平面。这是等周与密铺共同优化的结果，高效又省材料。
Among tilings, hexagons enclose the most area per wall length and tile the plane without gaps. It is the joint optimum of isoperimetry and tessellation — efficient and material-saving.
难度 入门 · 适合 大众
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
只用一种正多边形能铺满平面的有哪几种？
Which regular polygons alone can tile the plane?
只有正三角形、正方形和正六边形三种，因为它们的内角能整除360度。正五边形不行，会在拼缝处留空隙。
Only the equilateral triangle, square and regular hexagon work alone, because their interior angles divide 360°. The regular pentagon leaves gaps.
难度 入门 · 适合 小学低段(6-8)
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
#镶嵌 #平面