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Thread: mathematics of patterns

  1. #1 mathematics of patterns 
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    There's a tendency for the evolution of mathematics to branch. If we assume the basic line of mathematics to be basic arithmetic, then one of the earliest branchings would have to be algebra. More recent developments might include calculus, combinations and permutations, trigonometry, set theory, logic and computer science, and so on. One thing I'm curious about is whether there is a mathematics of patterns. You know, repetitions and regularities. Is there such a thing?


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    I'm not aware of a general branch that deals with patterns. Due to the very nature of mathematics, I'd say pattern is not confined to a specific branch but is rather ubiquitous in the entire subject. As Wikipedia says concerning pattern: mathematics is (sometimes called) the "Science of Pattern".

    But there may be some abstract branch dealing with order theory and pattern analysis that I'm not aware of, so stay tuned for following answers from other members. What I can do however is point you to the realm of fractals, which are patterns themselves with very interesting properties.


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    Forum Junior JoshuaL's Avatar
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    I think your right epidecus about the idea that math itself is the study of pattern, in all its different guises. Topics like tessallations are explored using multiple mathematical disciplines. Same with chaos theory. Some disciplines might be more along the lines of what you're thinking though. Check out complexity.
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    Moderator Moderator Markus Hanke's Avatar
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    Quote Originally Posted by epidecus View Post
    I'm not aware of a general branch that deals with patterns.
    Not patterns, but symmetries. The branch dealing with that would be group & representation theory.
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    Thanks, both, for your replies.
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    Thanks for posting your question. I believe with patterns we can learn to predict the future, discover new things and better understand the world around us. Many patterns we see around us have symmetry. Numbers can also have patterns and we can use algebra for describing those patterns.
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    Quote Originally Posted by epidecus View Post
    I'm not aware of a general branch that deals with patterns. Due to the very nature of mathematics, I'd say pattern is not confined to a specific branch but is rather ubiquitous in the entire subject. As Wikipedia says concerning pattern: mathematics is (sometimes called) the "Science of Pattern".

    But there may be some abstract branch dealing with order theory and pattern analysis that I'm not aware of, so stay tuned for following answers from other members. What I can do however is point you to the realm of fractals, which are patterns themselves with very interesting properties.
    As I understand it, CDT (causal dynamic triangulation) is one of the latest areas of inquiry in physics itself. It assumes a fundamental fractal spcetime geography.
    Causal Dynamical Triangulation: Booting Up The Space-Time Continuum - YouTube
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    Quote Originally Posted by Write4U View Post
    As I understand it, CDT (causal dynamic triangulation) is one of the latest areas of inquiry in physics itself. It assumes a fundamental fractal spcetime geography.
    Causal Dynamical Triangulation: Booting Up The Space-Time Continuum - YouTube
    Is that to say spacetime is fundamentally a fractal pattern?
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    Quote Originally Posted by epidecus View Post
    Quote Originally Posted by Write4U View Post
    As I understand it, CDT (causal dynamic triangulation) is one of the latest areas of inquiry in physics itself. It assumes a fundamental fractal spcetime geography.
    Causal Dynamical Triangulation: Booting Up The Space-Time Continuum - YouTube
    Is that to say spacetime is fundamentally a fractal pattern?
    As I understand it, a CDT universal geometry is not in conflict with either GR or QM, while offering a very hospitable geometry which allows for spacetime to exist as we know it.

    The beauty of fractals (we are surrounded by fractals everywhere) is their fundamental simplicity which can yield infinite complexity.

    from wiki,
    Images of fractals can be created by fractal generating programs.
    1) Iterated function systems – use fixed geometric replacement rules; may be stochastic or deterministic;[32] e.g., Koch snowflake, Cantor set, Haferman carpet[33], Sierpinski carpet, Sierpinski gasket, Peano curve, Harter-Heighway dragon curve, T-Square, Menger sponge
    2) Strange attractors – use iterations of a map or solutions of a system of initial-value differential equations that exhibit chaos (e.g., see multifractal image)
    3) L-systems - use string rewriting; may resemble branching patterns, such as in plants, biological cells (e.g., neurons and immune system cells[17]), blood vessels, pulmonary structure,[34] etc. (e.g., see Figure 5) or turtle graphics patterns such as space-filling curves and tilings
    4) Escape-time fractals – use a formula or recurrence relation at each point in a space (such as the complex plane); usually quasi-self-similar; also known as "orbit" fractals; e.g., the Mandelbrot set, Julia set, Burning Ship fractal, Nova fractal and Lyapunov fractal. The 2d vector fields that are generated by one or two iterations of escape-time formulae also give rise to a fractal form when points (or pixel data) are passed through this field repeatedly.
    5) Random fractals – use stochastic rules; e.g., Lévy flight, percolation clusters, self avoiding walks, fractal landscapes, trajectories of Brownian motion and the Brownian tree (i.e., dendritic fractals generated by modeling diffusion-limited aggregation or reaction-limited aggregation clusters).[6]
    Fractal - Wikipedia, the free encyclopedia

    http://www.bing.com/images/search?q=...ry&FORM=RESTAB
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    Quote Originally Posted by Write4U View Post

    As I understand it, CDT (causal dynamic triangulation) is one of the latest areas of inquiry in physics itself. It assumes a fundamental fractal spcetime geography.
    Causal Dynamical Triangulation: Booting Up The Space-Time Continuum - YouTube
    Actually, that isn't an assumption, but a consequence of CDT. The underlying assumption is that space-time is made up of discreet "building blocks" called simplexes, and if you follow this line of thought and do the maths it turns out that the vacuum on a microscopic level has fractal properties.
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    Quote Originally Posted by Markus Hanke View Post
    The underlying assumption is that space-time is made up of discreet "building blocks" called simplexes...
    Okay, let me see if I can wrap my head around the concept. So it's assuming space-time is merely an emergence of these varying simplexes?
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    Moderator Moderator Markus Hanke's Avatar
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    Quote Originally Posted by epidecus View Post
    Okay, let me see if I can wrap my head around the concept. So it's assuming space-time is merely an emergence of these varying simplexes?
    Yes precisely, it is the result of those simplexes arranging themselves in a certain way, thereby forming space-time.
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