关于亲吻圆织物的研究
On a Fabric of Kissing Circles
AI总结：
本文通过圆反演构建亲吻圆织物，证明其框架圆曲率为双无限等差数列、链圆曲率为二次双序列，并应用结果解决两个算额问题。
摘要：
将圆反演应用于一个填满圆的正方形网格，我们得到了一种我们称之为亲吻圆织物的构型。该构型及其组成部分——两个正交框架和两个正交的链族——在某种程度上与经典几何构型相关联，如阿基米德双圆（arbelos）或帕普斯链（Pappus chain），或20世纪的阿波罗尼奥斯填充（Apollonian packing）。在本文中，我们构建了这种织物，并列举了由这一构造产生的若干明显性质。接下来，我们关注各个组成部分内部的曲率：我们证明框架圆的曲率构成一个双无限等差数列（双序列），而每条链中的圆的曲率则排列成一个二次双序列。由于解决几何算额（sangaku）问题是我们发现这种织物的途径，我们在论文结尾给出了两个算额问题及其利用我们关于曲率的结果得到的解答。
英文摘要：
Applying circle inversion on a square grid filled with circles, we obtain a configuration that we call a fabric of kissing circles. The configuration and its components, which are two orthogonal frames and two orthogonal families of chains, are in some way connected to classical geometric configurations such as the arbelos or the Pappus chain, or the Apollonian packing from the 20th century. In this paper, we build the fabric and list some of the obvious properties that result from this construction. Next, we focus on the curvature inside the individual components: we show that the curvatures of the frame circles form a doubly infinite arithmetic sequence (bi-sequence), whereas the curvatures of the circles of each chain are arranged in a quadratic bi-sequence. Because solving geometric sangaku problems was a gateway to our discovery of the fabric, we conclude this paper with two sangaku problems and their solutions using our results on curvatures.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
