Category Archives: Mathematics

Recipe for Mathematics

April 3, 2019 Martin K. Jones 1 Comment

Guided only by their feelings for symmetry, simplicity, and generality, and an indefinable sense of the fitness of things, creative mathematicians now, as in the past, are inspired by the art of mathematics rather than by any prospect of ultimate usefullness.

— E. T. Bell

A smile fell on the grass.
Irretrievable!

And how will your night dances
Lose themselves. In mathematics?

— Sylvia Plath, from The Night Dances

Further Reading:

https://en.wikipedia.org/wiki/Eric_Temple_Bell

https://hellopoetry.com/poem/710/the-night-dances/

https://poetrywithmathematics.blogspot.com/

[*11.14]

eightfolds, fourfolds, Mathematics

Degen’s Eight-square Identity

March 25, 2019 Martin K. Jones Leave a comment

Here is another identity but this time corresponding to an eight-fold: the Eight-square identity of Ferdinand Degen found about 1818. You know the drill: it states that a product of two numbers that are each the sum of eight squares is itself the sum of eight squares!

(a₁² + a₂² + a₃² + a₄² + a₅² + a₆² + a₇² + a₈²)(b₁² + b₂² + b₃² + b₄² + b₅² + b₆² + b₇² + b₈²) =

…The sum of the expressions in the eight triangles written in the diagram above. (Please consult the Wikipedia entry below for the textual formulas, as it’s too hard to write in HTML.)

Note that the expressions above have an interesting symmetry, aside from the one on the upper left. Indeed, Euler’s Four-square Identity has a similar simpler symmetry. There is also a connection with Octonions if you are interested in digging for it. If you are anticipating that there is such a formula for sums of sixteen squares, there is, but not a bilinear one, and it is much more complicated!

Further Reading:

https://en.wikipedia.org/wiki/Degen%27s_eight-square_identity

https://en.wikipedia.org/wiki/Pfister%27s_sixteen-square_identity

[*11.64]

fourfolds, Mathematics

Euler’s Four-square Identity

March 24, 2019 Martin K. Jones 1 Comment

Here’s a nice little math identity that mathematician Leonard Euler wrote down in a letter dated 1748. It states that the product of two numbers that are sums of four squares is itself the sum of four squares.

(a₁²+a₂²+a₃²+a₄²)(b₁²+b₂²+b₃²+b₄²) =

(a₁b₁-a₂b₂-a₃b₃-a₄b₄)² + (a₁b₂+a₂b₁+a₃b₄-a₄b₃)² + (a₁b₃+a₃b₁+a₄b₂-a₂b₄)² + (a₁b₄+a₄b₁+a₂b₃-a₃b₂)²

It can be proved with elementary algebra or even by quaternions!

Further Reading:

https://en.wikipedia.org/wiki/Euler%27s_four-square_identity

[*11.62]

fourfolds, logic, Mathematics, sixteenfolds

Four Forms Make a Universe, Part 2

March 17, 2019 Martin K. Jones 1 Comment

This is a continuation of my last entry. Above is a different representation of the LICO alphabet, with the letters turned 45 degrees counter-clockwise, and rearranged into a symmetric pattern. The letters seem to arise more naturally in this orientation, but then Schmeikal rotates them into his normal schema.

And to the right is a diagram of the logical expressions that correspond to the letters above.

After making these new diagrams, I became inspired and made a few other figures to share with you.

These two versions, with triangles instead of line segments, and also with borders between adjacent triangles removed:

And these two versions, with quarter circles, and also with edges between adjacent quarter circles removed:

Further Reading:

Four Forms Make a Universe

[* 11.50, *11.58]

fourfolds, logic, Mathematics, sixteenfolds, Space and Time

Four Forms Make a Universe

March 12, 2019 Martin K. Jones 4 Comments

How could I not love a paper with this title? I’ve struggled with it for a bit, and I’ve only managed a couple of diagrams relating the author’s LICO (Linear Iconic) alphabet made up of 16 letters. However, I see that there are a few other papers by Schmeikal available on ResearchGate that look easier to understand. But also however, the first one says to read the “Four Forms” paper first!

At any rate, I present a sixteen-fold of the LICO alphabet, and another of the binary Boolean operators that are in a one-to-one mapping with LICO. There is much to understand from these papers, including much syncretism between various mathematical sixteen-folds, so please forgive me if I don’t explain it all with immediate ease. However, I believe it is well worth the effort to understand.

(Please note that the characters of the LICO alphabet are oriented so that the bottoms of the letters are downward, but the Boolean operators are oriented so that the bottoms of the equations are towards the right angles of the triangles.)

The title comes from the result that four elements of LICO can reproduce the other twelve via linear combinations. These four forms are 1) Boolean True (A or ~A), 2) A, 3) B, and 4) A=B. These are within the interior right-hand triangles in the LICO diagram. Of course, it is well known from Computer Science that the NAND operator (~A or ~B) can also generate all other fifteen operators, but this is by multiple nested operations instead of simple Boolean arithmetic. There are several other “universal” binary gates that can do this as well.

Two other representations that have four elements that can generate the other twelve via linear combinations come from CL(3,1), the Minkowski algebra. These representations are called “Idempotents” and “Colorspace vectors”. Because of this algebra’s association with space and time in relativity, Schmeikal claims that LICO has ramifications in many far-ranging conceptualizations.

Further Reading:

Bernd Schmeikal / Four Forms Make a Universe, in Advances in Applied Clifford Algebras (2015), Springer Basel (DOI 10.1007/s00006-015-0551-z)

https://link.springer.com/article/10.1007/s00006-015-0551-z

At http://www.researchgate.net:

Bernd Schmeikal / Free Linear Iconic Calculus – AlgLog Part 1: Adjunction, Disconfirmation and Multiplication Tables

Bernd Schmeikal / LICO a Reflexive Domain in Space-time (AlgLog Part 3)

https://en.wikipedia.org/wiki/Spacetime_algebra

https://www.allaboutcircuits.com/technical-articles/universal-logic-gates/

[*9.145, *11.50]

eightfolds, fourfolds, Mathematics

Octonions

February 25, 2019 Martin K. Jones 1 Comment

Octonions are the fourth, the last, and the greatest of the so-called normed division algebras based on the Real numbers. When I first learned about the Octonions way back when, I didn’t like them because they weren’t associative like the Quaternions, the Complex numbers, and the Reals. But now I’m fine with that, and they may be important for new theories of physics!

Octonions have the general form:

a₀e₀ + a₁e₁ + a₂e₂ + a₃e₃ + a₄e₄ + a₅e₅ + a₆e₆ + a₇e₇

Where the coefficients a_i are Real and the bases e_i have (something like) the following relations:

e₀ = 1 (and -1 = e₁² = e₂² = …)

e₁ = I = e₂e₃ = e₇e₆ = e₄e₅

e₂ = J = e₅e₇ = e₃e₁ = e₄e₆

e₃ = IJ = e₁e₂ = e₆e₅ = e₄e₇

e₄ = K = e₅e₁ = e₆e₂ = e₇e₃

e₅ = IK = e₇e₂ = e₁e₄ = e₃e₆

e₆ = JK =e₅e₃ = e₁e₇ = e₂e₄

e₇ = IJK = e₆e₁ = e₃e₄ = e₂e₅

In addition, if any of the products like e₂e₃ = e₁ are reversed you get the negative, so e₃e₂ = -e₁.

Non-associativity is demonstrated by going through the list of triples:

(e₁e₂)e₃ = e₃² = -1

e₁(e₂e₃) = e₁² = -1

(e₁e₂)e₄ = e₃e₄ = e₇

e₁(e₂e₄) = e₁e₆ = -e₇

(e₁e₂)e₅ = e₃e₅ = -e₆

e₁(e₂e₅) = e₁e₇ = e₆

so it is hit or miss I guess. Also note that e₇ = (IJ)K = -I(JK). And for all (e_ie_j)e_k and e_i(e_je_k), if they are not equal, is one equal to the negative of the other? And do I have to multiply them all out to find out?

Further Reading:

https://en.wikipedia.org/wiki/Octonion

http://math.ucr.edu/home/baez/octonions/

https://www.quantamagazine.org/the-octonion-math-that-could-underpin-physics-20180720/

http://mathworld.wolfram.com/Octonion.html

https://ncatlab.org/nlab/show/normed+division+algebra

[*11.33]

fourfolds, Mathematics

Quaternions

February 22, 2019 Martin K. Jones 1 Comment

The Quaternions are a number system that enlarges the Complex numbers, just as the Complex numbers enlarge the Real numbers. In fact, Quaternions can be thought as special pairs of Complex numbers, just as Complex numbers can be thought as special pairs of Real numbers.

Quaternions can be used for all sorts of wonderful things, such as rotations in 3D space, instead of using matrices. Above is a pitiful diagram (although better than my last one) of the Quaternion units 1, i, j, and k used in the typical representation a + b i + c j + d k. Please read about them in the links below and be amazed!

Further Reading:

https://en.wikipedia.org/wiki/Quaternion

https://en.wikipedia.org/wiki/Quaternions_and_spatial_rotation

http://mathworld.wolfram.com/Quaternion.html

http://www.euclideanspace.com/maths/algebra/realNormedAlgebra/quaternions/

https://www.quantamagazine.org/the-strange-numbers-that-birthed-modern-algebra-20180906/

[*10.160]

fourfolds, Mathematics, Science

The Free Energy Principle

November 27, 2018 Martin K. Jones Leave a comment

Below is a link to a fascinating article about Karl Friston, whose research on the “free energy principle” (also known as active inference) tries to explain how biological (or even artificial) systems maintain or even increase their organization. Without much explanation, here are some details. The model is a system with four main variables:

Sense (s)
Action (a)
Internal States (r or μ)
Hidden States (ψ)

Sense and action divide the internal states of the system from the hidden states external to it. This division is called a “Markov Blanket”. I’ve tried to show the equations between the variables correctly but I’m not too sure if I have them right. They seem to change from paper to paper. F is an expression of the free energy, but I’m not sure what f is at the moment, except for being some sort of “flow”.

Further Reading:

https://www.wired.com/story/karl-friston-free-energy-principle-artificial-intelligence/

https://en.wikipedia.org/wiki/Free_energy_principle

https://en.wikipedia.org/wiki/Markov_blanket

http://rsif.royalsocietypublishing.org/content/15/138/20170792

[*11.6]

fourfolds, Mathematics

Viete’s Method of Constructing Pythagorean Triples

October 24, 2018 Martin K. Jones Leave a comment

Number is the ruler of forms and ideas, and the cause of gods and daemons.

— Pythagoras, as attributed by Iamblichus

This diagram shows (but not to scale ;-)) a clever method of constructing two Pythagorean triples given any two other Pythagorean triples due to Francois Viete.

Given (a, b, c, d, e, f are integers):

a^2 + b^2 = c^2
d^2 + e^2 = f^2

two other triples are found by synaeresis:

A = ae + bd
B = be – ad

and diaeresis:

D = ae – bd
E = be + ad

(I guess a, b, d, and e can always be picked so that B > 0 and D > 0, that is, be > ad and ae > bd?)

so that:

A^2 + B^2 = (cf)^2
D^2 + E^2 = (cf)^2

Voila!

Further Reading:

https://en.wikipedia.org/wiki/Pythagorean_triple

Special thanks to Pat’s Blog for this little gem:

https://pballew.blogspot.com/2018/10/viete-on-pythagorean-triples.html

Additionally, these relationships are also used in Fermat’s Theorem on sums of 2 squares

https://en.wikipedia.org/wiki/Proofs_of_Fermat%27s_theorem_on_sums_of_two_squares

[*9.138, *10.186, *10.187]

fourfolds, Mathematics, Science, technology

Science, Technology, Engineering, and Mathematics

June 14, 2018 Martin K. Jones Leave a comment

STEM: Science, Technology, Engineering, and Mathematics. We often hear that these areas of education and expertise are critical for the development of our modern society. To attract students to these fields, banners and logos are full of bright colors and crisp graphics. In comparison, above is my rather dull diagram. Not very enticing, is it?