(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.
- Penrose Tiles to Trapdoor Ciphers by Martin Gardner
- Algebraic Theory of Penrose’s Non-periodic Tiligns of the Plane by N.G. de Bruijn
- The Empire Problem in Penrose Tilings by Laura Effinger-Dean
Image Source: Wikipedia
The gif animation of folding the net of the tesseract(8-cell). ;-)
There are many kinds of the net of the tesseract.
I think “Geometrical Constructions” is a handy reference about geometry.
In figure 25: Draw a circle that will tangent two lines and go through a given point C on the line F C, which bisects the angle of the lines.Through C draw AB at right angles to C F; bisect the angles D A B and E B A, and the crossing on C F is the center of the required circle.
Or In figure 28: To plot out a circle arc without recourse to its center, but its chord A B and height h being given.
With the chord as radius, and A and B as centers, draw the dotted circle ares A C and B D. Through the point 0 draw the lines A O o and B O o, Make the arcs C o = A o and D o = B o. Divide these arcs into any desired number of equal parts, and number them as shown on the illustration. Join A and B with the divisions, and the crossings of equal numbers are points in the circle arc.
A happy number is a number defined by the following process: Starting with any positive integer, replace the number by the sum of the squares of its digits, and repeat the process until the number equals 1 (where it will stay), or it loops endlessly in a cycle which does not include 1. Those numbers for which this process ends in 1 are happy numbers, while those that do not end in 1 are unhappy numbers. More formally, given a number n = n_0, define a sequence n_1, n_2,… where n_(i+1) is the sum of the squares of the digits of n_i. Then n is happy if and only if there exists i such that n_i = 1. If a number is happy, then all members of its sequence are happy; if a number is unhappy, all members of the sequence are unhappy.
Especially, the happiness of a number is unaffected by rearranging the digits, and by inserting or removing any number of zeros anywhere in the number. For example, 19 is happy, as the associated sequence is:
1^2 + 9^2 = 82
8^2 + 2^2 = 68
6^2 + 8^2 = 100
1^2 + 0^2 + 0^2 = 1.
There are infinitely many happy numbers and infinitely many unhappy numbers. Consider the following proof: 1 is a happy number, and for every n, 10n is happy since its sum is 1, and for every n, 2 × 10n is unhappy since its sum is 4 and 4 is an unhappy number.
It is possible to define a consistent addition of points on certain kinds of curves (elliptic curves). This arithmetic plays an important role in modern mathematics. For instance, Wiles’ proof of Fermat’s last theorem is a consequence of the modularity theorem (once known as the Taniyama-Shimura-Weil conjecture), which gives a strong connection between elliptic curves and modular forms. Elliptic curves over finite fields also have cryptographic applications, or can be used for integer factorization.
Four Coins Problem: Given three coins of possibly different sizes which are arranged so that each is tangent to the other two, find the coin which is tangent to the other three coins.
A special case of Apollonius’ problem requiring the determination of a circle touching three mutually tangent circles (also called the kissing circles problem). There are two solutions: a small circle surrounded by the three original circles, and a large circle surrounding the original three. Frederick Soddy gave the formula for finding the radius of the so-called inner and outer Soddy circles given the radii of the other three. The relationship is
(k1 +k2 + k3 +k4)^2 = 2.[(k1)^2 +(k2)^2 + (k3)^2 +(k4)^2]
(1/r1 +1/r2 + 1/r3 +1/r4)^2 = 2.[ (1/r1)^2 +(1/r2)^2 + (1/r3)^2 +(1/r4)^2]
The outer Soddy circle and inner Soddy circle are the solutions to the four coins problem.
- Method to construct basic 3 tangent circles & Outer - Inner Soddy circle.
Construction methods based primarily on the properties of the (interior) bisector of an angle of the triangle. The circle is the set of all points in a plane that are at a given distance from a given point and is a highly symmetric shape.
- Here is method construct basic 3 tangent circles: The key is to first construct a triangle. Pick the three points that you want to be the centers of the three circles. These points will also be the vertices of the triangle. For this triangle, construct the incenter. From the incenter, construct a perpendicular line through the incenter to each of the sides. This is also how to construct the incircle. The intersection points of the perpendicular line to each side of the triangle is going to be the point where any two circles are tangent. We have now constructed the three tangent circles.
- References: Method to construct Outer & Inner Soddy circle at Jwilson.coe.uga.edu/SoddyCircles.
See more at: Outer Soddy circle , Inner Soddy circle & Descartesn circle theorem on Mathworld.wolfram.com. - http://en.wikipedia.org/wiki/Descartes%27_theorem - http://en.wikipedia.org/wiki/Isoperimetric_point.
Mathematical chess problems: Knight’s tour & Longest uncrossed knight’s path.
- A knight’s tour is a sequence of moves of a knight on a chessboard such that the knight visits every square only once. If the knight ends on a square that is one knight’s move from the beginning square (so that it could tour the board again immediately, following the same path), the tour is closed, otherwise it is open. The knight’s tour problem is the mathematical problem of finding a knight’s tour. Variations of the knight’s tour problem involve chessboards of different sizes than the usual 8 × 8, as well as irregular (non-rectangular) boards.
- The longest uncrossed (or nonintersecting) knight’s path is a mathematical problem involving a knight on the standard 8×8 chessboard or, more generally, on a square n×n board. The problem is to find the longest path the knight can take on the given board, such that the path does not intersect itself. A further distinction can be made between a closed path, which ends on the same field as where it begins, and an open path, which ends on a different field from where it begins.