# symmetry Published articles for symmetry. This is one page of public article previews, not the complete archive. Follow Next page to continue. Summaries are not the original full articles. ## The Aboulafia Graph Has Dihedral Symmetry DevFeed: [The Aboulafia Graph Has Dihedral Symmetry]() Original publisher: [Read original article]() Author: Yehonathan Sharvit Published: 2026-07-06T10:00:00Z Content type: article Language: en Sources: [Klipse]() Topics: [Graphs]() Tags: [aboulafia](), [graph](), [math](), [mirror](), [rotation](), [series](), [symmetry]() ### AI overview The final article in a four-part series explains the dihedral symmetry of the Aboulafia graph. It connects the graph's recursive construction to a rotation and a reflection, and describes a proof that these are its only symmetries. ### Source excerpt Aboulafia's Tserouf - Part 4 of 4 <- Previous: Too big to draw, but yet drawable ## Two's Complement and Group Theory DevFeed: [Two's Complement and Group Theory]() Original publisher: [Read original article]() Published: 2023-07-10T07:00:00Z Content type: article Language: en Sources: [Jeremy Kun]() Topics: [math](), [Mathematics](), [Computer science](), [Programming]() Tags: [abelian-groups](), [arithmetic](), [bits](), [boolean](), [circuits](), [computer-science](), [group-actions](), [groups](), [math](), [mathematics](), [programming](), [quotients](), [symmetry](), [twos-complement]() ### AI overview The article explains two's-complement signed integer arithmetic using group theory. It presents signed and unsigned n-bit integers as representations of the quotient group of integers modulo 2^n, clarifying why the same arithmetic circuits can operate on both. ### Source excerpt Before I discovered math, I was a first year undergrad computer science student taking Electrical Engineering 101. The first topic I learned was what bits and boolean gates are, and the second was the two's complement representation of a negative n-bit integer. At the time two's complement seemed to me like a bizarre quirk of computer programming, with minutiae you just had to memorize. If the leading bit is 1, it's negative, and otherwise it's positive. ## A Rook Game DevFeed: [A Rook Game]() Original publisher: [Read original article]() Published: 2014-08-31T17:51:37Z Content type: tutorial Language: en Sources: [Jeremy Kun]() Topics: [Algorithm](), [Mathematics]() Tags: [algorithm](), [board-games](), [chessboard](), [game](), [games](), [mathematics](), [strategy](), [symmetry]() ### AI overview This article presents a rook game on an 8x8 chessboard in which players move only south or west. It shows that symmetry provides the optimal strategy: move the rook to the diagonal whenever possible and return it there after the opponent moves it away. The first player wins from an off-diagonal starting square; otherwise, the second player wins. ### Source excerpt Problem: Two players take turns moving a rook on an 8x8 chessboard. The rook is only allowed to move south or west (but not both in a single turn), and may move any number of squares in the chosen direction on a turn. The loser is the player who first cannot move the rook. What is the optimal play for any starting position? rook-board Solution: Take advantage of the symmetry of the board. ## Groups -- A Primer DevFeed: [Groups -- A Primer]() Original publisher: [Read original article]() Published: 2012-12-08T23:42:10Z Content type: article Language: en Sources: [Jeremy Kun]() Topics: [Mathematics](), [structure](), [Math and Logic](), [Cryptography]() Tags: [applications](), [group-actions](), [groups](), [lagrange](), [mathematics](), [public-key](), [set-theory](), [structure](), [symmetry](), [theory]() ### AI overview This primer introduces group theory as the study of algebraic structures that can describe operations beyond ordinary numbers. It motivates the subject through examples involving symmetry, squares, and Rubik's Cube, and notes applications including public-key cryptography, error detection, crystal structures, particle physics, and mathematical biology. ### Source excerpt The study of groups is often one's first foray into advanced mathematics. In the naivete of set theory one develops tools for describing basic objects, and through a first run at analysis one develops a certain dexterity for manipulating symbols and definitions. But it is not until the study of groups that one must step back and inspect the larger picture. The main point of that picture (and indeed the main point of a group) is that algebraic structure can be found in the most unalgebraic of settings. ## Metric Spaces -- A Primer DevFeed: [Metric Spaces -- A Primer]() Original publisher: [Read original article]() Published: 2012-08-26T12:08:25Z Content type: tutorial Language: en Sources: [Jeremy Kun]() Topics: [Mathematics](), [Math and Logic](), [Algorithms, Complexity]() Tags: [algorithms](), [levenshtein-distance](), [mathematics](), [metric](), [symmetry](), [topology](), [triangle-inequality]() ### AI overview This tutorial introduces metric spaces as sets equipped with distance functions that satisfy non-negativity, symmetry, and the triangle inequality. It explains that metrics impose a topology and contrasts metric-based work with approaches using non-metrics. ### Source excerpt The Blessing of Distance We have often mentioned the idea of a "metric" on this blog, and we briefly described a formal definition for it. Colloquially, a metric is simply the mathematical notion of a distance function, with certain well-behaved properties. Since we're now starting to cover a few more metrics (and things which are distinctly not metrics) in the context of machine learning algorithms, we find it pertinent to lay out the definition once again, discuss some implications, and explore a few basic examples. ## Inner Product Spaces--A Primer DevFeed: [Inner Product Spaces--A Primer]() Original publisher: [Read original article]() Published: 2011-07-25T00:29:00Z Content type: tutorial Language: en Sources: [Jeremy Kun]() Topics: [Mathematics](), [Math and Logic](), [Computing]() Tags: [big-o-notation](), [eigenvectors](), [inner-product](), [mathematics](), [matrices](), [orthogonality](), [symmetry](), [vector-spaces](), [vectors]() ### AI overview This primer explains why vector spaces can be extended with a dot-product-like operation and defines inner products through conjugate symmetry, linearity, additivity, and positive definiteness. It introduces inner product spaces and discusses their connection to linear functionals. ### Source excerpt Vector spaces alone are not enough to do a lot of the interesting things we'd like them to do. Since a vector space is a generalization of Euclidean space, it is natural for us to investigate more specific types of vector spaces which are more akin to Euclidean space. In particular, we want to include the notion of a dot product. By admitting additional structure to a vector space, we may perform more computations, and hopefully get more interesting results.