Talk:Involution (mathematics)

message: I removed this from the main page as it has not context. We need to say who believes this, what religion (if any) these beliefs are a part of, etc etc. Probably beter if it lives in its own article.
(As developed by Sri Aurobindo:)
The process by which the Divine manifest the cosmos is called the Involution. The process by which that which was created rises to higher states and states of consciousness is the evolution. The Involution is essentially up to the point of the Big Bang; the Evolution is from that point forward.
 
After the creation, the Divine (i.e. the Absolute, Brahman, God; all these essentially mean the same thing) is both the One (the Creator) and the Many (that which was created).
--
The process by which the Many is created from out the Absolute is called the Involution. Once that process ends, the process of Evolution begins. In essence the evolution begins with creation (the Big Bang being a stage) and continues with all that follows. 
 
The involution is the process by which the Absolute manifest the creation, the universe. It is the process by which the Many emerged from the One as a universe of divided, ignorant forms. The involution is that which occurred that enabled the creation, the universe, the cosmos to manifest from out of the Original Principle, the Divine, God, the Absolute. Involution is the process of self-limitation, of densification, by which the Absolute, Brahman veils itself by stages until it assumes the appearance in the cosmos, the universe we know of. It wishes to create the universe to objectify itself and its spiritual properties in infinite possibilities, for the purpose of delight of discovery which it will achieve thereafter.
 
The evolution is the movement forward by which the created universe evolves from its initial state of divided, ignorant forms, emerges as Life and Mind, and in that process rediscovers its Source. The evolution occurs after the involution. It is the development and progressive movement of all in the cosmos, including humans, to attain its fulfillment, including rediscovery in delight of the spiritual aspect, that Consciousness-Force, that was the source of the creation. The evolution is the progressive development from the first inconscience in matter into life (movement, sensation, etc. and living physical beings), to mind (in conscious being, animals, including the human, the self-conscious thinking animal), to spiritualized mind, culminating in Supermind, Truth Consciousness (as supramental individuals, leading to a supramental, i.e. a divine life on earth.)
DJ Clayworth 15:28, 26 Jan 2004 (UTC)
thread_title: Religious stuff (not math)

message: This statement is false:
: A permutation is an involution precisely if it is a transposition.
(1 2)(3 4) is an involution, but it is not a transposition (at least according to the definition on the Wikipedia transposition page).  I've changed it. -- Walt Pohl 21:20, 7 Dec 2004 (UTC)
thread_title:  Transpositions 

message: This disambiguation notice is obviously wrong:
: 
: This article is about involution in mathematics. For other uses, see Involution (mathematics) (disambiguation).
: 
If I were familiar with the templates, I'd fix it. Michael Hardy (talk) 12:48, 4 September 2008 (UTC)
thread_title:  Incorrect dab notice 

message: Should the exclusive-or operator be mentioned? Given an arbitrary value x, the operation f(a) = a xor x is its own inverse. In particular, g(a) = a xor 0 is the identity function. | Loadmaster (talk) 19:45, 4 February 2009 (UTC)
thread_title:  Boolean exclusive-or 

message: This page has an absurd reference. The paper referenced in the sources section, "Quaternion involutions and anti-involutions", regurgitates mathematics that has been known since the time of William Hamilton. That paper should not have even been published. Use a serious reference, such as "The book of involutions" (http://www.math.uni-bielefeld.de/~rost/BoI.html). This is a comprehensive text written by leaders in algebra. Or if you're looking for something more accessible, why not go for a paper of John Voight (of U. Vermont)? He's the current world expert on the study of quaternion algebras. I could think of dozens of more appropriate references.  — Preceding unsigned comment added by 128.84.79.145 (talk) 16:15, 16 September 2011 (UTC)
thread_title:  Ridiculous reference 

message: Is there a name for the subset of involution functions used in most reciprocal ciphers? i.e.,
: The function pairs up each element x with some other element y, such that
: f(x) = y and x != y and f(y) = x, for each and every x.
Or in other words,
: f(x) != x for each and every x, and
: f(f(x)) == x for each and every x
(In particular, the identity function is not a member of this subset).
My understanding is that the Enigma machine and nearly all other reciprocal ciphers always use such a function to transform each plaintext letter to the corresponding ciphertext letter.
(I suspect because of the misunderstanding that "when we encrypt a plaintext letter, obviously the encrypted ciphertext should not be the same letter").
I've been calling such a function a pairing, but I've recently discovered that term is usually used to mean something quite different -- what term should I use instead?
--DavidCary (talk) 18:35, 26 November 2013 (UTC)
thread_title:  The specific kind of involution used in most reciprocal ciphers 

message: The following text was removed:
:For  in ℝ, this is often called Babbage's functional equation (1820). Ref: 
The function f(x) = 1 – x is indeed an involution, but the article cited is about more general periodic functions and is inappropriate here. Perhaps another editor can find a place for it elsewhere.Rgdboer (talk) 23:10, 28 June 2015 (UTC)
thread_title:  Babbage 

message: I think the examples should be the general cases of -x, and 1/x (to be C-x and C/x). I also think that it is a bit misleading to show that the composition of the above functions is also an involution, cause the reader might think that any composition of two involutions is also involution. besides, composition of the above function, can be considered as the mere one case of the general case, which is C/x (where C=-1) 2A02:6680:1102:385C:1841:9A2D:D966:A308 (talk) 20:04, 7 June 2022 (UTC)
thread_title:  Please give the reader the general case 

message: All given examples for involutions in the reel numbers can be created with the composition of a bijection g with an involution f with the formula g \circ f \circ g^{-1}:
e.g.:  f_3 (x)=f_2 (x-1)+1  and  f_4(x) = \sqrt{2} f_3(x/\sqrt{2}) . A special case is a shift along the line y=x.
Is it possible to reduce all involutions to a composition starting with the inverse operations of the reell numbers  -x , \frac{1}{x} ? Conscious 'n' curious (talk) 11:34, 18 November 2022 (UTC)
thread_title:  Generel Properties