Whatever happened to humility?

Archive/RSS/Ask/Theme





Steven. 21. Dallas-TX. Uni-Senior. Gay. Dr. Pepper. Rum raisin ice-cream. Optimistically realistic. Humility. Instagram: @Stevenhle

ΠΚΦ, ΙΞ #24

Science. Medicine. Art. Music. Funny. Scary. Horror. Witty. Morality. Existentialism. Culinary arts. Ethics. Atheism. Passion. Empathy. World. Humanity. Steven.

A Penrose tiling can be constructed using just two different titles in the shape of a thick and thin rhombus:

Here, the angles are multiples of ø = π/5. The edges of these tiles are marked with two different kinds of arrows. This is done to enforce a specific matching rule that ensures that the tiling is non-periodic, which is one of the defining features of Penrose tilings. Different tiles can only be placed next to each other if the touching edges have the same type of arrow and point in the same direction.

If these matching conditions were not in place, then it would be possible to construct tilings which are periodic as shown in the following image. There are translational symmetries present in 8 different directions here:

The image at the top shows a part of a Penrose tiling obeying the matching conditions with the arrows displayed. The numbers are just there to index some other property of directionality not discussed here.

If these matching conditions are satisfied for a complete tiling, then the resulting configuration will always be non-periodic. However, following the matching rules alone does not guarantee an infinite tiling of the entire plane. It is therefore possible to construct finite regions that obey the matching rules, but cannot be extended any further without allowing for periodicities or contradicting the matching rules.

There do exist sets of tiles that will always admit non-periodic tilings in which no extra matching conditions need to be imposed. For a list of such tiles that tile the plane, 3-dimensional space, and even the hyperbolic plane, see this list of aperiodic sets of tiles.

Image sources:

(via proofmathisbeautiful)

  1. vangoghld reblogged this from callmeklimt
  2. clark-ed reblogged this from callmeklimt
  3. callmeklimt reblogged this from geometryofdopeness
  4. chowyuheil reblogged this from davidhannafordmitchell
  5. whcottonsiu reblogged this from intothecontinuum
  6. happyvsright reblogged this from leahmaths
  7. kithandkin reblogged this from spring-of-mathematics
  8. rapidmentaldecline reblogged this from spring-of-mathematics
  9. imperfect-aspect reblogged this from vostok1
  10. hiddenline reblogged this from spring-of-mathematics
  11. leahmaths reblogged this from spring-of-mathematics
  12. spring-of-mathematics reblogged this from geometryofdopeness
  13. emotionsasdata reblogged this from geometryofdopeness and added:
    A Penrose tiling can be constructed using just two different tiles in the shape of a thick and thin rhombus: Here, the...
  14. geometryofdopeness reblogged this from laukitsch
  15. ksssssssshk reblogged this from ngngngngngngng
  16. wildoute reblogged this from ngngngngngngng
  17. mmmxii reblogged this from ngngngngngngng
  18. ngngngngngngng reblogged this from laukitsch