(click through the images to view in high-res)

Penrose tilings are an example of the non-periodic tilings discussed in the last post. Recall that these are tilings that cover the entire infinite plane leaving neither gaps nor overlaps. Whats nice about these tilings is that the set of tiles used to construct the Penrose tilings only consists of two different basic shapes consisting of quadrilaterals

Whats even more remarkable about Penrose tilings is that using just these two shapes it is possible to construct infinitely many different tilings that cover the infinite plane. This infinity is infinitely bigger in size than the countable infinity of the whole numbers ( 1, 2, 3, 4, and so on), but rather equivalent to the uncountable infinity associated to the real numbers (which includes all whole numbers, fractions, and decimals with infinitely many digits).

Despite the existence of these infinitely many different Penrose tilings, there is one peculiar property they all exhibit. Consider any finite region, or patch, of one particular Penrose tiling. Then it is possible to find an exact copy of this patch in any other different Penrose tiling! Moreover, this patch occurs infinitely often in different spots in any given tiling! Note that this is true of any patch of any size no matter how big as long as its finite. This implies that if you were only able to examine a finite part of any Penrose tiling, you could never really distinguish that entire tiling from any other tiling. Thus, different Penrose tilings are only perfectly distinguishable in the infinite limit of the entire plane.

The images shown above display finite regions of Penrose tilings. They are constructed using an elegant “cut-and-project” method (also used here), which involves projecting the points of the integer lattice in 5-dimensional Euclidean space, onto a certain 2-dimensional plane. Connecting adjacent projected points in this plane by lines then yields a Penrose tiling.

Image Source: Wikipedia

1. yanagiken reblogged this from intothecontinuum
2. selfesteempunk reblogged this from corngrind
3. corngrind reblogged this from elotromonte
4. elotromonte reblogged this from yuruyurau
5. yuruyurau reblogged this from mirandamolina
6. eliason reblogged this from intothecontinuum and added:
penrose tiling is THE BEST.
7. empiricallyoptimistic reblogged this from intothecontinuum
8. missingrache reblogged this from gessorly and added:
Excuse me what is a trapdoor cipher also these are gorgeous I like it when math and art do things together, someone...
9. gessorly reblogged this from roachpatrol
10. harrietlane reblogged this from intothecontinuum
11. origamybills reblogged this from intothecontinuum
12. fromvictory reblogged this from splitfailed and added:
this is some crazy “Mandelbrot Set” shit right here.
13. splitfailed reblogged this from shiny1
14. shiny1 reblogged this from faeofterra
15. warningpageexpired reblogged this from intothecontinuum
16. justalittlerandomer reblogged this from pyrrhiccomedy
17. patterntrip reblogged this from reallifeprodigy
18. faeofterra reblogged this from cicadahusks
19. robotnoise reblogged this from pyrrhiccomedy
20. sophia-sol reblogged this from pyrrhiccomedy
21. cicadahusks reblogged this from intothecontinuum
22. cephalopodink reblogged this from roachpatrol
23. autumninganymede reblogged this from phasmidhugs
24. lifeiscarrots reblogged this from roachpatrol
25. hotdog-apocalypse reblogged this from roachpatrol
26. olanthanide reblogged this from pyrrhiccomedy and added:
1) This is awesome. 2) This also reminds me of my high school math teacher singing “aleph null bottles of beer on the...
27. 8bitian reblogged this from roachpatrol
28. brsis reblogged this from pyrrhiccomedy
29. jzumun reblogged this from spinor
30. amayafire reblogged this from roachpatrol
31. jellyfishdirigible reblogged this from roachpatrol
32. bootslots reblogged this from pyrrhiccomedy
33. phasmidhugs reblogged this from intothecontinuum