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Math

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Andrius Kulikauskas

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See: Math

Christopher Alexander 15 Transformations

Christopher Alexander's principles of life and different kinds of opposites are relevant.

Main points:

  • Centers arise from the "center".
  • Centers (and structures) are preserved.

Look for:

  • The role of duality...
  • The role of opening up ever new dimensions...
  • Where we imagine the center - beyond us or among us. In our plane or beyond it?
  • The sense of time - Mandelbrot set is given by the time for each point to settle down.
  • The interaction with us - the simplex engages us, our imagination, in a way that the Mandelbrot set does not.
  • Relate Mandelbrot set to the kneading process.

Levels of scale. Perturbations of a center A yield centers B, one level of scale smaller. The subcenters are first latent and then cause a nucleation around A. The level of scale is what is needed to preserve (the symmetry of) the existing structure. Injects more beautifully articulated intermediate levels of scale. (A process of clearing out and preserving at different levels of scale, alternating activity and structure at different levels.) New parts are similar in size to one another, but one level smaller than the center which they differentiate. The smaller parts make the large center more coherent and distinct.

Strong centers. New centers by their position and arrangement strengthen existing centers. Most important transformation. Weak center is made more emphatic, more strongly differentiated, defined, integrated by virtue of its differences, more sharply drawn and distinguished. Gives weight and definition and distinction to any center which has begun to crystallize in any given field.

Boundaries. Boundary zone occurs where the steepest gradient of differentiation of the centers drops off around the center. As the boundary grows intenser it fosters centers within the boundary. The boundary fills with activity. First, a cloudy but distinctly differentiated zone with some "character" appears. It is further differentiated, becoming a thick, distinct boundary.

Alternating repetition. Repeating centers caused by similar conditions themselves create similar conditions which establish alternate centers. They accentuate each other. A repeating pattern of similar centers is paired with a second, alternate pattern of repeating centers, interlocking and alternating with the first. Thus a large system is given a structure as a repeating field of many repeating smaller entries.

Positive space. Empty space tends to become better defined around latent centers, more pronounced and positive. Centers balance against each other. New centers are created in the space between other centers, thus strengthening and shaping spaces between the other centers that are not yet centers themselves.

Good shape. Vaguely existing centers are replaced by definite centers, strengthening the shape. Intensifies the products of alternating repetition, strengthening them, making them more distinctive, along with the positive shape transformation. Any loosely formed shape is made more marked, stronger, beautiful, living, giving strength to the centers within it.

Local symmetries. Symmetrical or near-symmetrical evolution maintains global structures and cleans up debris. Some centers are refinforced by this. The density of local symmetries increases. Injects local symmetry into weaker, emerging centers. Makes each center in the system an internal axis of symmetry. Symmetrizes the kernel of the center. Gives strength to an emerging center. It makes a newly differentiated identity a strong center in its own right.

Deep interlock and ambiguity. Random perturbations form disturbances at latent centers. Centers go in one direction or another, one zone or another along the edge. Edge centers can belong to the larger centers on both sides of the line. Minor centers along the edge swell in size and penetrate more deeply into the zones of the two large centers on either side of the edge. (The minor centers perturb into the scale above them and disturb it.) They thus interprenetrate the larger centers and cause deep interlock. When the centers may belong to either one side or the other we get spatial ambiguity. Weaves the opposite sides of the boundary into a tighter, less separated union by physically creating connections where one part enters into the other and vice versa, thus cementing the whole.

Contrast. Each center may be made more distinct, differentiated from its surroundings, through oppositions of contrasting polarities. Differences increase and forms stand out more sharply. Both internal differentiation and differentiation from the surrounding environment are increased. Increases the distinction between two types of center, separating them more sharply from each other by way of some polarity. Creates a well-knit system where the two kinds of centers complement each other better.

Gradients. The intensification of the center, the reorganization of space around the center, yields (inverse-square) gradients oriented toward the center, strengthening it by a field effect. Creates transitions of size and character. Introduces systematic variation in an uneven or non-homogeneous field. Introduces coherence in a random-like structure. Thus creates structure where none was visible before. Or a simple polarity, position or axis engenders a gradient, and the inner parts and centers are given features which vary systematically, making perhaps a global effect. Orders complex and inchoate structure without greatly bending or changing circumstance.

Roughness. As a system comes to order, the increasing structure-preserving pressure to form and support the largest centers will refine smaller features to subtly (though inexplainably) adapt them to the globalities. Irregularities allow the global pressures to be resolved. Uses intentional irregularity to find the most regular fit possible to make things work out in the large.

Echoes. Similarities of process create similar systems of centers. Creates a widespread family resemblance which provides unity.

The void. Highly differentiated structure is set off against an empty, clean, smooth structure to overcome confusion and crowding. Irrelevant structure is cleaned out. Gets rid of garbage. Areas which are relatively undifferentiated are cleaned out and made more homogeneous and defined by a boundary zone which is attached, surrounded by more differentiated structure. Preserves an imitation of the greater undifferentiated void.

Simplicity and inner calm. Nothing unnecessary remains present. Irrelevant and confusing centers have been removed. Essential structure is allowed to remain. Cleans, simplifies, by removing unwanted centers, differences and other kinds of complexity in disparate places. Reduces unnecessary structure.

Not-separateness. Wrapping each center into a web of other centers. The pressure to unify and tie everything together. Create an unbroken tissue of organisms. Wed each part more firmly to the others. Eliminate exaggerated differences. Place small infill centers for fine tuning. Modifies centers and their surroundings so that the center gains more subtlety of its surroundings and the surroundings gain more substance from the center, bringing the two closer together, forming an indissoluble unity. Knits together to create a texture, reduces the separateness of any given entity. Takes pieces of A and copies them inside B, and takes pieces of B and copies them inside A. Thus making A and B more associated, more allied, more united and less distinct from one another. Binds a being and its surroundings, a center and its context, more tightly. Makes them more connected, similar, different, interlocked, complementary, reminiscent of each other, both more and less distinct, more united. Makes inside and outside less distinguishable.


Matematikos grožis

Kas gražu matematikoje ir kodėl?

One milestone (or miracle) in knowing everything is to generate all mathematical objects and truths in a unified way, but especially, generate all mathematical insights in a way that builds our intuition. The relevant outlook will, I imagine, be in terms of beauty. I mean that we may think of mathematical insight as guided by the wish for beauty.

Beauty - wholeness preserving transformations

  • natural generalizations
  • coordinate free

Mathematical beauty

  • Wikipedia: In the 1990s, Jürgen Schmidhuber formulated a mathematical theory of observer-dependent subjective beauty based on algorithmic information theory: the most beautiful objects among subjectively comparable objects have short algorithmic descriptions (i.e., Kolmogorov complexity) relative to what the observer already knows.[17][18][19] Schmidhuber explicitly distinguishes between beautiful and interesting. The latter corresponds to the first derivative of subjectively perceived beauty: the observer continually tries to improve the predictability and compressibility of the observations by discovering regularities such as repetitions and symmetries and fractal self-similarity. Whenever the observer's learning process (possibly a predictive artificial neural network) leads to improved data compression such that the observation sequence can be described by fewer bits than before, the temporary interestingness of the data corresponds to the compression progress, and is proportional to the observer's internal curiosity reward.
  • Math Beauty blog and Readings
  • What is Good Mathematics? Terrence Tao.
  • Alexander Bogomolny, Cut the Knot Manifesto "The peculiar beauty of Mathematics lies in deduction, in the dependency of one fact upon another. The less expected a dependency is, the simpler the facts on which the deduction is based -- the more beautiful is the result."
  • Beautiful, simple proofs

Mathematicians are often guided by their sense of beauty. Beauty is a key to the big picture in mathematics as well as physics and philosophy.

Questions about beauty

  • What is meant by beauty?
  • What principles determine it?
  • To what extent is beauty objective and subjective?
  • How does beauty lead to mathematical insight?

Investigation: Collect examples

In this section, we collect examples of beauty in mathematics.

Richard Stanley’s post at Math Overflow, Most intricate and most beautiful structures in mathematics, produced the following list, ordered by votes received:

  • The absolute Galois group of the rationals
  • The natural numbers (and variations)
  • Homotopy groups of spheres
  • The Mandelbrot set
  • The Littlewood Richardson coefficients (representations of Sn etc.)
  • The class of ordinals
  • The monster vertex algebra
  • Classical Hopf fibration
  • Exotic Lie groups (See the Scientific American article about A.Garrett Lisi’s application of E8 to physics: A Geometric Theory of Everything)
  • The Cantor set
  • The 24 dimensional packing of unit spheres with kissing number 196560 (related to the monster vertex algebra).
  • The simplicial symmetric sphere spectrum
  • F_un (whatever it is)
  • The Grothendiek-Teichmuller tower.
  • Riemann’s zeta function
  • Schwartz space of functions

Below we gather more examples, both basic and advanced:

  • e^2pii + 1 = 0
  • more generally, Euler’s formula: e^2piix = cos(x) + i sin(x)
  • the unique decomposition of natural numbers into prime numbers
  • Euler’s polyhedron formula V - E + F = 2
  • the classification of the Platonic solids
  • the relationship between a polynomial and its graph
  • binomial theorem and Pascal's triangle
  • elementary proofs of the Pythagorean theorem (such as: "four times a right triangle is the difference of two squares" - the basis for infinitesimal rotation).

Investigation: Analyze examples

In this section, we analyze the examples collected above to consider:

  • In what sense are they beautiful?
  • What makes them beautiful?
  • What are the simplest examples of beauty?
  • Which examples yield the most beauty for the least drudgery?

Investigation: Look for unifying principles or contexts.

  • Urs Schreiber notes that many of the beautiful structures relate to string theory.
  • Relation between two completely different domains, especially dual, complementary domains.

Investigation: Compare with beauty in chess.


Matematikos grožis žadina vaizduotę

Tęsiu prieš dvejus metus skaitytą pranešimą apie kūrybos prasmę ir meno taisykles. Žavesio jausmą nusakysiu kaip paneigimą bjauresio sąlygų - vidinio gyvenimo. Užtat grožis yra tiesos atskleidimas išoriniame gyvenime. Dviem pavyzdžiais iš matematikos parodysiu kaip architekto Kristoferio Aleksandro gyvybės dėsniai įvairiais kampais nuduoda arba išsako kūrybinę prasmę. Neišsemiamas fraktalas Mandelbrot aibė magina akis, tačiau matematiko vaizduotę žadina simpleksas. Paprastojo simplekso paveika grįsiu dėsnius, kaip grožis žadina vaizduotę. Paaiškinsiu, kaip Mandelbrot aibė galėtų labiau sužavėti matematiką.

1) Gražuolė tampa vis gražesnė. 2) Nebylė vs. tūkstantis šypsenų. Grožis: išsisakymas: Dvasios trilypė meilė visiems: išsakytojas, išdėstymas, klausytojas.

Akis (1 taškas); šypsena (1 tiesinė atkarpa) - tai Mona Lisa šypsena.

Supažindinti su dviem pavyzdžiais Mandelbrot ir simplex, ir parodyti kaip jie tampa gražūs. Abiems pavyzdžiams pritaikyti Christopher Alexander gyvybės dėsnių lentelę, kurią prieš dvejus metus pristačiau.

Simplex yra:

  • total order
  • decomposition
  • all six pertvarkymai?

Opposites:

  • square roots of -1
  • 1 and -1
  • point and center

Simplekse ryšį tarp taško ir židinio galime ištrinti - galime ištrinti tašką - ir tuo suglaudinti sandarą - ir tai yra gražu. Gražu, kad židinys nepažymėtas.

Sieti su Christopher Alexander mintimis.

Mandelbrot aibė nėra matematikui graži nes tai yra gražu akims o ne vaizduotei.

Mandelbrot eiga irgi nėra graži nes tai tarsi atsitiktinė.

Tačiau ji tampa graži, kada paaiškėja

  • jog tai išplaukia iš begalinės sekos
  • jog tai nusakoma Catalan skaičių
  • jog tai skaičiuoja kompiuterių generuojamą kalbą
  • jog tai gali būti užkoduota kiekviename taške kompleksinių skaičių plokštumoje

Harvey Friedman nurodė kokie ypatingi yra įrodymai, kad kažkas neįmanoma:

  • impossible to square the circle
  • the real numbers are not countable
  • Goedel's incompleteness theorem

But these are not beautiful... because they are negative results, not constructive results.

There are other results which are amazing because they go beyond contemporary axiomatics:

  • imaginary numbers
  • Euler's tricks with divergent infinite series, the Riemann function
  • infinitesimals
  • sphere of radius i
  • Ramanujan

These are pretty in some sense, but not beautiful. There is a sense of a method at work, but it is not grounded, not systematic.

So what is beautiful is something between these extremes. It is a framework between the two which can ground the two types of results. What is beautiful is when the imagination can "see" the gist in either case.

Penrose https://www.researchgate.net/publication/247981405_The_role_of_aesthetics_in_pure_and_applied_mathematical_research_Bull_Inst_Math_Appl

http://www.kurzweilai.net/how-bio-inspired-deep-learning-keeps-winning-competitions

Sąmoningumas - savasties ženklas - nes savastis kartojasi - nuo kurios gali būti skirtumai būti paskaičiuoti.

Suglaustinumas - compressibility - means a pattern exists - Christopher Alexander's recurrent activity evokes structure. (+2) Search for new patterns. Self-compression - want to compress oneself as a pattern compressor. (Consciousness +3) "Interesting" = derivative of beauty.

Juokai remiasi glaudinimu.


P.S. I forgot to say in my other letter that I'll be suggesting a talk at a philosophy (aesthetics) conference here in Vilnius, Lithuania, most likely to be accepted, about mathematical beauty. I appreciate thoughts on mathematical beauty (I suppose through a new thread). My main thought so far is that mathematicians (at least me) would typically not consider the Mandelbrot set as beautiful because you can only see it, you can't imagine it. Whereas Galois theory is beautiful because it empowers the imagination. So I want to explain what it takes for the Mandelbrot set to become beautiful for a mathematician. Also, I want to link in with architect Christopher Alexander's 15 principles of life.


I'll bring up some thoughts that may be relevant for you and your art teachers and art students. I'm thinking through a talk that I will propose for an aesthetics conference here in Vilnius, Lithuania. I want to talk about mathematical beauty. So I'm wondering what I can say about that. But I'm thinking it would be good to use my work with the Mandelbrot set as an example.

I think my main idea is that, from a mathematician's point of view, (at least my own), the Mandelbrot set is not beautiful. It's not beautiful because in math beauty is not what you see, but what you imagine. And I can't imagine the Mandelbrot set. It's just lots of noise. There's no melody. Whereas I can imagine an equilateral triangle and so it stands out amongst all of the triangles. I can play with it in my mind, watch it dance around. I can hum that tune.

What's truly beautiful is Galois theory where you have this assurance that you can play around with a group of dynamic actions and that will correspond to a polynomial and its solutions. That is amazingly beautiful. There is order in the universe. Or the dualities that you notice with the Platonic solids. Or the fact that there can only be those solids and no others, that what matters are the number of edges and vertices and faces and whether they satisfy Euler's formula. That makes it seem like we have signs of a Why out there some where.

But I will go through steps to show what can make the Mandelbrot set beautiful mathematically, step by step. The kinds of steps I wrote about that this is not some accidental set. And that it relates to lots of key things in math. And that (perhaps) every single point is actually encoding something meaningful, so that the whole complex plane is an analysis of all of the possible behaviors of automata. And furthermore if the intricate structure of the Mandelbrot set was visually displaying how those behaviors are related. Something like that would be awesomely beautiful.

MathBeauty


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Puslapis paskutinį kartą pakeistas 2016 rugsėjo 05 d., 08:31
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