AI技术的魅力:数学公式画成美丽的图像!

如果你看过吴军老师的《数学之美》，那么那些数学定理公式的美是抽象的，在想象中的，但现在借助AI画图！

附图中推特用户Tivadar Danka（twitter.com/TivadarDanka）用数学定理在Midjourney中画的图片。

图2. Baire类别定理： "在一个完整的度量空间中，无数个密集集的交集仍然是密集的。"

The Baire category theorem: "In a complete metric space, the intersection of countably many dense sets remains dense."

图3. 佐恩定理："一个包含每条链的上限的部分有序集合必然包含至少一个最大元素"。

Zorn's lemma: "A partially ordered set containing upper bounds for every chain necessarily contains at least one maximal element."

图4. 微积分基本定理："一个函数的导数的积分可以恢复原来的函数，直到一个常数"。

The fundamental theorem of calculus: "The integral of a function's derivative recovers the original function, up to a constant."

图5. 巴纳-塔斯基悖论："将一个实心球体分解成有限个不相交的子集，然后将这些子集重新组合，形成两个与原球体相同的球体。"

The Banach-Tarski paradox: "Decomposing a solid sphere into a finite number of disjoint subsets, and then reassembling those subsets to create two spheres identical to the original one."

图6. "每个向量空间都有一个哈默尔基"。

"Every vector space has a Hamel basis."

图7. 代数基本定理："每个非常数多项式方程至少有一个复数根"。

The fundamental theorem of algebra: "Every non - constant polynomial equation has at least one complex root."

图8. 哥德尔不完备性定理："在任何形式化的公理系统中，都有无法在系统内证明的真实语句，系统的一致性也无法由其自身的公理来证明"。

G?del's incompleteness theorems: "In any formal system of axioms, there are true statements that cannot be proven within the system and the consistency of the system cannot be proven by its own axioms."

图9. 算术基本定理："每个大于1的正整数都可以唯一地表示为素数的乘积"。

The fundamental theorem of arithmetic: "Every positive integer greater than 1 can be represented uniquely as a product of prime numbers."

图10. Brouwer的固定点定理： "在欧几里得空间中一个紧凑的凸集的任何连续变换中，至少有一个点保持固定"。

Brouwer's fixed point theorem: "In any continuous transformation of a compact, convex set in Euclidean space, there is at least one point that remains fixed."

图11. 中心极限定理： "大量独立且相同分布的随机变量之和将近似于正态分布，而不考虑原始分布。"

The central limit theorem: "The sum of a large number of independent and identically distributed random variables will be approximately normally distributed, regardless of the original distribution."

图12. 海涅-博莱尔定理： "欧几里得空间的紧凑子集正是那些封闭和有界的子集。"

The Heine-Borel theorem: "The compact subsets of Euclidean space are precisely those that are closed and bounded."

图13. 奇异值分解： "每个矩阵都可以被分解为一个单元矩阵、一个对角线矩阵和另一个单元矩阵的乘积"。

The singular value decomposition: "Every matrix can be decomposed into the product of a unitary, a diagonal, and another unitary matrix."

图14. 奖励："实数的集合是不可计数的无限的，以萨尔瓦多-达利的风格。"

Bonus: "The set of real numbers is uncountably infinite, in the style of Salvador Dali."

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