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    Dave Horner's Website - Yet another perspective on things...
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    $$\cos x = \sum\limits_{n = 0}^\infty {\frac{{\left( { - 1} \right)^n x^{2n} }}{{\left( {2n} \right)!}}}$$

    penrose tribar, the impossible infinite triangle

    Sunday, 24 June 2018 07:56

    penrose tribar, the impossible triangle

    celebrated as a paradox or a unique contradiction, a mystery that can’t be solved...
    a triangle that looks like a real, solid three-dimensional object, but isn't.
    The impossible triangle, infinite triangle, penrose tribar, impossible tribar, is a triangular impossible object.
    a continuous loop, an ambiguous figure, a visual paradox;
    a strange loop with tangled hierarchies.
    impossibility in its purest form

    looks like a simple object. 2d depiction of a 3d triangle built from square beams. The eyes travel from face to face of the object, everything looks great, no problem. As the viewer traverses the continuous impossible object their perspective flips back and forth between equally possible perspectives in an unconscious and automatic way.

    implicit in this strange loop is the concept of infinity. a loop representing an endless process in a finite way. -a conflict between the finite and the infinite.

    helps give a sense that perceptions are limited and subjective from those of another person viewing the same thing.

    A "penrose" polygon refers to the 2-dimensional depiction of the impossible object itself.
    Penrose triangle - Wikipedia
    Roger Penrose was the inventor of the tribar.
    it's also an interesting look at Escher.
    video with the Roger Penrose talking about the discovery of the tribar (spanish).
    Escher's works like ascending decending, waterfall, and other tilings are influenced by the Penrose father and son's work.
    non-isohedral tilings, tiles only one shape but they have different roles to play in different parts of the pattern. Roger sent Escher a wooden non-isohedral block puzzle to Escher; and Escher's solution was in Escher's final print "Ghosts" which he made just before he died. Roger says in the video "Unfortunately, Escher died before the non-periodic tilings that came somewhat later."


    Tessellations - M.C. Escher Ghosts - Escher's last tessellation "Ghosts" was a solution to a puzzle sent to him by Roger Penrose, the mathematician. Escher solved it and, true to form, changed the angular wood blocks into rounded 'ghosts'.


    if a line is traced around the Penrose triangle, a 4-loop Möbius strip is formed....

    and ya, that's my leg and first tattoo. and ya, I'm white and nerdy. Thank you Samuel Hyndman


    60 Penrose Triangle Tattoo Designs For Men - Impossible Tribar Ideas -
    Triangle Tattoo Meaning | herinterest.com/
    Impossibility on Behance
    Making a Penrose Triangle on Behance - make a penrose perspective paper and tape model.
    If you're interested in optical illusions, you might also checkout Optical tricks and illusions - Dave Horner's Website.

    should maybe be on another penrose page but...

    Tilings
    Penrose tiling - Wikipedia
    Aperiodic tiling - Wikipedia
    Penrose Tiles -- from Wolfram MathWorld
    Impossible Cookware and Other Triumphs of the Penrose Tile - Issue 13: Symmetry - Nautilus
    Anisohedral tiling - Wikipedia - In geometry, a shape is said to be anisohedral if it admits a tiling, but no such tiling is isohedral (tile-transitive); that is, in any tiling by that shape there are two tiles that are not equivalent under any symmetry of the tiling. A tiling by an anisohedral tile is referred to as an anisohedral tiling.
    Isohedral figure - Wikipedia -isohedral or face-transitive when all its faces are the same. More specifically, all faces must be not merely congruent but must be transitive, i.e. must lie within the same symmetry orbit. In other words, for any faces A and B, there must be a symmetry of the entire solid by rotations and reflections that maps A onto B. For this reason, convex isohedral polyhedra are the shapes that will make fair dice.
    Penrose diagram - Wikipedia - is a two-dimensional diagram capturing the causal relations between different points in spacetime. It is an extension of a Minkowski diagram where the vertical dimension represents time, and the horizontal dimension represents space, and slanted lines at an angle of 45° correspond to light rays.
    Orchestrated objective reduction (Penrose–Lucas argument) - Wikipedia - The Penrose–Lucas argument states that, because humans are capable of knowing the truth of Gödel-unprovable statements, human thought is necessarily non-computable.


    AMS :: Feature Column :: The Topology of Impossible Spaces
    The Penrose Tribar(pdf)
    Drawing A Tribar Figure by Hand(pdf)
    AMS :: Feature Column :: The Topology of Impossible Spaces - this page includes amazing pictures and walks you through the math behind perspective and how to know if an object is impossible in mathematical terms. 1-dimensional cohomology
    Impossible Triangle - this page right here has wonderful english to walk you through what's going on in a tribar, i highly recommend it. "The beams of the triangle simultaneously appear to recede and come toward you. Yet, somehow, they meet in an impossible configuration!" "it is possible to construct a physical model of the impossible triangle that looks impossible from only one angle" "It would lead to biological disaster if you were blind to the unusual, inconsistent, or paradoxical." "When you examine a drawing of an impossible triangle, you first build up a global image by examing the local parts." "When you examine a drawing of an impossible triangle, you first build up a global image by examing the local parts."
    Impossible Triangle - The Illusions Index - another good page with wonderful english description of impossible triangle. It also includes a four sided impossible figure.

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    Last Updated on Thursday, 09 August 2018 15:06