Symmetry

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Symmetry (an often should be presented in its plural form symmetries, as a namespace or configuration space) is a term connected to the ideas of Invariance, Equivalence, Reversible logic, and Conservation. On Page 180 of Lawvere's book[1], Felix Klein of Klein bottle suggested a way to study objects using symmetries. Sir William Hamilton also had a one-pager memo that relates quaternion with the subject of symmetry[2]. This idea was later discussed in the paper on General Theory of Natural Equivalence as the foundational paper[3] of Category Theory.

Symmetries as the first Meta-Rule

According to Mathemaniac, symmetries can be thought of as mathematical operands that gets to be manipulated through some operations that preserves the properties of being symmetrical. These four most general properties are:

  1. Closure: Symmetrical operations on symmetries always create symmetries
  2. Associativity: Symmetries composition with symmetries are symmetries Associative
  3. Identity/Unit: Doing nothing is a symmetrical operation
  4. Inverse Exists: Symmetrical operations can be undone, and returns to the original symmetry.

A mathematical treatment of this subject was explained by Norm Wilberger in a video[4].


Given the meta-rule about symmetry, one may consider applying these properties to the manipulation of functions. More specifically, one may utilize some functional programming language to automate the transformation operations to manipulate functions.

Functions as symmetrical objects

This can be implemented using a combination of JavaScript, Cascading Style Sheets, and HTML as a combination of functional programming language, declarative rule engine, and a display mark-up rendering formatting specification language. It would be particularly convenient to manage the interactions of these three kinds of languages using MediaWiki's existing infrastructure.


A few excellent tutorial videos on Symmetry so far

Cheung, Trevor (Jul 9, 2021). Chapter 1: Symmetries, Groups and Actions - Essence of Group Theory. local page: Mathemaniac. 


Galois Theory provides a computational framework for studying symmetry. |Ct2fyigNgPY}}


References

  1. ↑ Lawvere, William; Schanuel, Stephen (January 8, 2009). Conceptual Mathematics_A First Introduction to Categories (2nd ed.). local page: Cambridge University Press. p. 180. ISBN 978-0521719162. 
  2. ↑ Sir William Rowan Hamilton (1856). "Memorandum respecting a new System of Roots of Unity" (PDF). Philosophical Magazine. local page. 12: 446. 
  3. ↑ Paper/General Theory of Natural Equivalence
  4. ↑ Wildberger, Norman J. (Nov 24, 2021). A (somewhat) new paradigm for mathematics and physics. local page: Insights into Mathematics. 

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