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When searching through the Oeis, I came across something called a floretion. Based on the context, it seems to be some sort of algebraic structure. I googled it and found nothing that explained their exact definition or motivation. Could someone explain its motivation and definition to me? A source would also be helpful if it explains the motivation.

The sequence is [A105163]: http://oeis.org/A105163

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A rather comprehensive collection of information on floretions, specifically in the context of Oeis, is Sequences related to floretions.

In essence, most of the "floretion" sequences come from an iterated function that begins with some initial value and produces one integer at each step of the iteration. The floretion algorithms operate on a 16-element quantity that can be thought of a a $4\times 4$ matrix. Such a 16-element quantity is called a floretion. The name is deliberately similar to quaternion, octonion and sedenion because the operations performed with them are similar. (But why doesn't the floretion have its own Wikipedia page?)

If you want to "hear" a floretion, here you go.

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    $\begingroup$ "Florenion" would be more similar to "quaternion" etc.. $\endgroup$ – Qfwfq Jun 30 '15 at 22:01
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The article Structure of the Floretion Group by Richard J. Mathar defines the Floretion Group as the group of order $32$ and GAP catalogue number $49$.

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The floretion website has remained totally unchanged for the last six years- this was not due to lack of interest, but more because of my complete change of professions. As of today, I've created a new dokuwiki extending floretions from 2nd (the way they were originally defined) to 4th order, meaning there are now 256 "base vectors" instead of 16. This is all just in the infant stages and can still only be serviced in my free time.

floretion dokuwiki (Wayback Machine)

Floretions themselves are defined in several places (click on the 2nd order tab in the wiki), so I will not go into detail again here. In summary, if Q = {+-i, +-j, +-k, +-e} is the quaternion group, a "base" floretion (of which there are 16) can be written ij, ik, ei, ej, ek, ie, je, ke, ... Examples:

ij * ik = -ei
ij * ee = ij

Upon extending to 4th order floretions, we have 256 base "super floretions" which can be written eeIJ, eiIJ, jkEJ, ..., where eeEE is the unit. Here, the first two letters are always lowercase, next two uppercase. Of course, this notation is not strictly necessary, but since the beginning I've found it helps to read from an optical standpoint.

As for the group itself, where 2nd order floretions are found to be isomorph to the factor space F = Q x Q\{(1,1), (-1,-1)}, 4th order floretions are isomorph to F x F\{(ee,ee),(-ee,-ee)} where ee is the unit. This group has 512 elements in total. Note in general when a "4th order floretion" is mentioned, we are not actually referring to the group itself, but to the 4th order floretion algebra over the reals, i.e. some vector

X = a*eeEE + b*eiEE + c*ejEE + ... with real coefficients a, b, c

Example multiplication of base vectors:

ijEK * jeKJ = <(ij*je) | (ek*kj)> = < kj | -ki > = -kjKI where ij, je, ek, kj are floretions. The complete multiplication table is given at the above link.

My main motivation is currently to find an equation similar to "Floret's Equation" for 2nd order floretions and to see what form "visual media" takes using both old and new algorithms on 4th order floretions. Currently, I've only included some files for multiplication in the project folder (written in R). However, I have begun to calculate basic algorithms and should be able to release those files very soon (I've also figured out how to "swap" elements but this needs some testing). I would be much obliged to any code reviewers / contributors.

Cheers, Creigh

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Updating this thread after approx. 5 years:

Edit: In the definition, below, we have $x∗y$ (disregarding signs) is given by bitFlip(bitXor(x, y), bitWidth=n*3) where x and y are any floretions of order n. Of course, any exact definition should take signs into account. Here is the code to properly multiply two floretions in C++ taking signs into account:

Let a and b be base floretions (i.e. integers with all digits equal 1, 2, 4, or 7 when written in octal) of order n (= the total number of digits in octal). Define

const long Oct666 = 0666666666666666666666;  
const long Oct111 = 0111111111111111111111;
// bitmask is unit floretion (0777...777) with number of digits = n 
long bitmask = pow(8, n) - 1; 

Then $a*b = $

int floret_ion::f_mult(int a, int b, const unsigned int bitmask) {

    int pre_sign = sgn<int>(a) * sgn<int>(b);

    a = abs(a);
    b = abs(b);

    // Shift every 3-bits of "a" one to the left; 
    unsigned int a_cyc = ((a << 1) & OCT666) | ((a >> 2) & OCT111);

    //take bit AND of a_cyc with b and determine if number of bits set is even or odd 
    int cyc_sign = (count_bits((a_cyc & b) & bitmask) & 0b1) ? 1 : -1;

    // take floretion order into account (e.g. "ije" is odd while "eeii" is even) 
    int ord_sign = (count_bits(bitmask) & 0b1) ? 1 : -1;


    return (pre_sign * cyc_sign * ord_sign) * (bitmask & (~(a ^ b)));

}

Original message:

There is a request for Floretion in WIKI going back to 2006, still pending. As one of the original contributors, I have collected a bunch of information, see below. However, I'm not a mathematician and would appreciate any collaboration.

A floretion is any whole number whose absolute value base 8 representation contains only digits 1, 2, 4 and 7. For example, the absolute value of the number -315337 in octal is 1147711, which contains only digits 1, 2, 4 and 7. The number of digits when written in octal is the order of the floretion (in this case 7). Instead of writing in octal, we could also associate $1 \leftrightarrow i$, $2 \leftrightarrow j$, $4 \leftrightarrow k$ and $7 \leftrightarrow e$ and write $iikeeii$, establishing a direct link to quaternions. Floretions of the same order can equipped with multiplication: $x*y$ (disregarding signs) is given by bitFlip(bitXor(x, y), bitWidth=n*3) where $x$ and $y$ are any floretions of order $n$. Signs are taken into account via the left bitshift operator (omitted here).

Floretions of the same order can be multiplied by scalars and added together, effectively turning the group of positive "base" vectors into an algebra over the reals. For example if $x = ie + .5jj + ki$ and $y = je + ik$, then $x*y = ke - .5ej - ek - ii - kk - ki$. Floretions of order one are just the quaternions. Multiplying floretions often leads to relationships among sequences involving fibonacci numbers.

Floretion groups are groups of order $2^{(2n+1)} = 2*4^n$ for $n = 1,2,3 \dots$ (there are $4^n$ positive base vectors)

Note all references to floretions before 2012 are of order 1 or 2. Links:

  1. Current definition
  2. Structure of Floretion Group, R. Mathar (order 2 only)
  3. OEIS A108618
  4. Construction of sequence with musical properties
  5. Online Floretion Multiplier (order 2 only)

  6. OEIS Sequences related to floretions

Additional info so far

Floretions can be used to generate algorithmic music, create fractals, show relationships between fibonacci numbers, store and process images, neural networks, and demonstrate aspects of necklace theory. This article discusses a general definition and uses basic image processing as an example. When floretions are discussed as groups, they are always finite. As each finite group has a matrix representation, broad areas of application are to be expected due to the wide range of matrix theory applications. Refer to the links for other examples.

Before proceeding, possible areas of confusion:

  1. Reading the word "quaternion" by itself, one may ask if the reference is to members of a group Q = {+-i, +-j, +-k, +-e}, to a ring, or to an algebra over the reals or over some other field. The same ambiguity occurs with "floretion": a reference may be to "some floretion x" without explicitely stating group, ring or algebra. This is only possible when the case is clear from the context (which it usually is). In most real-world applications, the reference is to the algebra over the reals. Here, a typical element may be written x = .5i - 2j + .75k or x = ii - ji + .5ee - jk. In the first case x is a first order floretion spanned by base floretions i, j, k (or 1, 2, 4 in octal notation) in the latter it is a second order floretion spanned by base floretions ii, ji, ee, jk (11, 21, 77, 24 in octal). Note floretions of different order cannot be added together or multiplied as they are not in the same space. As a consequence of the above possibilities, there are also many different ways to represent floretions notationally. Do we want to express an order 2 floretion as a member of the quaternion factor space QxQ{1,-1}, as a member of the tensor product of HxH where H is the real algebra of quaternions, as a 4x4 matrix of real numbers, in binary/decimal/octal, or in "OEIS notation" as x = ii - .5ji - ke? Here we will restrict ourselves to binary/decimal/octal or OEIS notation.

  2. The word 'order' is not used in the conventional sense when discussing floretions. Generally, when referring to the order of a floretion x, the reference is not to the standard definition "smallest m such that a^m = e" (for base floretions, m is always 1, 2 or 4) . A (base) floretion can be written with digits 1, 2, 4, 7 as 1224, 224, 12777. The order of the floretion is simply the number of digits. The group of order n floretions is a group with 2*4^n elements (the factor 2 corresponding to positive and negative elements). As the size of the group is generally also referred to the order of the group, there is a chance of confusion. Again, this should be clear from the context. For example, simply reading "order m" with no additional info, we already know the reference must be to an order m floretion from a group with 2*4^m elements, provided m cannot be written in the form 2*4^n for some n > 0. Conversely, reading "order 32" without any additional info is almost certainly referring to a group with 32 elements: simply because calculating properties floretions of order 12 and above (in this case 32) would be extremely time consuming: we would be working with groups with 2*4^(12) total elements! The only real chance of confusion the "order 8" case- but here we can agree to just say "quaternions".

An image processing introduction:

A floretion can be visualized as a heightmap above and/or below a two-dimensional square grid surface. Alternatively, if heights are interpreted as grayscales, a floretion can be seen as a square black and white image (clearly, 3 floretions as a color image). The higher the order of the floretion, the greater the grid resolution: order 1: 4^1 tiles, resolution 2x2 pixels order 2: 4^2 tiles, resolution 4x4 pixels order 3: 4^3 tiles, resolution 8x8 pixels (chessboard) ...

This applies whether we are discussing floretion groups or algebras: in the case of groups, all heights are equal 1 or -1. A key observation is that multiplication is defined, so these height maps and images can be manipulated in many ways. For example, we may store an image "A" as a floretion (a one dimensional sequence of numbers which can easily be converted to a two dimensional image) and require turning it by 90 degrees. Instead of defining the standard rotation matrix, we can look for an image B such that A*B gives the appropriate rotation. We could also look for a floretion with an inverse to distort the original image and restore it later.

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For more information on floretions, see here: http://www.fumba.eu/sitelayout/floretion.html

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  • $\begingroup$ It's better to give more information in your answer, not just a link, even if useful. $\endgroup$ – Alex Degtyarev Jul 30 '15 at 10:24
  • $\begingroup$ Yeah, it's called writing from work :) $\endgroup$ – Blackbeard Jul 30 '15 at 10:29
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    $\begingroup$ Actually, Carlo already gave that link in his answer on June 30. $\endgroup$ – Joe Silverman Jul 30 '15 at 15:03
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    $\begingroup$ The link is broken. $\endgroup$ – YCor Feb 1 at 7:41

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