Wang tile - Wikipedia Square tiles with a color on each edge The basic question about a set of Wang tiles is whether it can tile the plane or not, i.e., whether an entire infinite plane can be filled this way. The next question is whether this can be done in a periodic pattern. Domino problem[edit] In 1961, Wang conjectured that if a finite set of Wang tiles can tile the plane, then there also exists a periodic tiling, which, mathematically, is a tiling that is invariant under translations by vectors in a 2-dimensional lattice. This can be likened to the periodic tiling in a wallpaper pattern, where the overall pattern is a repetition of some smaller pattern. The Domino Problem deals with the class of all domino sets. In other words, the domino problem asks whether there is an effective procedure that correctly settles the problem for all given domino sets. Aperiodic sets of tiles[edit] The smallest set of aperiodic tiles was discovered by Emmanuel Jeandel and Michael Rao in 2015, with 11 tiles and 4 colors.
Platonic Solid -- from Wolfram MathWorld The Platonic solids, also called the regular solids or regular polyhedra, are convex polyhedra with equivalent faces composed of congruent convex regular polygons. There are exactly five such solids (Steinhaus 1999, pp. 252-256): the cube, dodecahedron, icosahedron, octahedron, and tetrahedron, as was proved by Euclid in the last proposition of the Elements. The Platonic solids are sometimes also called "cosmic figures" (Cromwell 1997), although this term is sometimes used to refer collectively to both the Platonic solids and Kepler-Poinsot solids (Coxeter 1973). The Platonic solids were known to the ancient Greeks, and were described by Plato in his Timaeus ca. 350 BC. In this work, Plato equated the tetrahedron with the "element" fire, the cube with earth, the icosahedron with water, the octahedron with air, and the dodecahedron with the stuff of which the constellations and heavens were made (Cromwell 1997). If 1. all lie on a sphere. 2. 3. 4. 5. Let (sometimes denoted (or , edges . . and
Mangahigh.com - Play maths, love maths Music Math Harmony -- Math Fun Facts It is a remarkable(!) coincidence that 27/12 is very close to 3/2. Why? Harmony occurs in music when two pitches vibrate at frequencies in small integer ratios. For instance, the notes of middle C and high C sound good together (concordant) because the latter has TWICE the frequency of the former. Well, almost! In the 16th century the popular method for tuning a piano was to a just-toned scale. So, the equal-tempered scale (in common use today), popularized by Bach, sets out to "even out" the badness by making the frequency ratios the same between all 12 notes of the chromatic scale (the white and the black keys on a piano). So to divide the ratio 2:1 from high C to middle C into 12 equal parts, we need to make the ratios between successive note frequencies 21/12:1. What a harmonious coincidence! The Math Behind the Fact: It is possible that our octave might be divided into something other than 12 equal parts if the above coincidence were not true!
Odd Numbers in Pascal's Triangle -- Math Fun Facts Pascal's Triangle has many surprising patterns and properties. For instance, we can ask: "how many odd numbers are in row N of Pascal's Triangle?" For rows 0, 1, ..., 20, we count: row N: 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 odd #s: 1 2 2 4 2 4 4 8 2 4 04 08 04 08 08 16 02 04 04 08 04 It appears the answer is always a power of 2. In fact, the following is true: THEOREM: The number of odd entries in row N of Pascal's Triangle is 2 raised to the number of 1's in the binary expansion of N. Presentation Suggestions: Prior to the class, have the students try to discover the pattern for themselves, either in HW or in group investigation. The Math Behind the Fact: Our proof makes use of the binomial theorem and modular arithmetic. (1+x)N = SUMk=0 to N (N CHOOSE k) xk. If we reduce the coefficients mod 2, then it's easy to show by induction on N that for N >= 0, (1+x)2^N = (1+x2^N) [mod 2]. Thus: (1+x)10 = (1+x)8 (1+x)2 = (1+x8)(1+x2) = 1 + x2 + x8 + x10 [mod 2].
Sums of Two Squares Ways -- Math Fun Facts In the Fun Fact Sums of Two Squares, we've seen which numbers can be written as the sum of two squares. For instance, 11 cannot, but 13 can (as 32+22). A related question, with a surprising answer, is: on average, how many ways can a number can be written as the sum of two squares? We should clarify what we mean by average. Let W(N) is the number of ways to write N as the sum of two squares. So if A(N) is the average of the numbers W(1), W(2), ..., W(N), then A(N) is the average number of ways the first N numbers can be written as the sum of two squares. A surprising fact is that this limit exists, and it is Pi! Presentation Suggestions: This might be presented after a discussion of lattice points in Pick's Theorem. The Math Behind the Fact: The proof is as neat as the result! Counting the the number of lattice points inside a circle is known as Gauss' circle problem. How to Cite this Page: Su, Francis E., et al.
A Random Math Fun Fact! From the Fun Fact files, here is a Random Fun Fact, at the Advanced level: The traditional proof that the square root of 2 is irrational (attributed to Pythagoras) depends on understanding facts about the divisibility of the integers. (It is often covered in calculus courses and begins by assuming Sqrt[2]=x/y where x/y is in smallest terms, then concludes that both x and y are even, a contradiction. See the Hardy and Wright reference.) But the proof we're about to see (from the Landau reference) requires only an understanding of the ordering of the real numbers! Proof. So, suppose Sqrt[2]=x/y, that is, x2 = 2y2; then we show x1 = 2y - x, y1 = x - y works. x/y = (2y - x) / (x - y). So x1/y1 yields the same fraction as x/y. Secondly, it must be the case that 0 < y1 < y, because this is the same as y < x < 2y, which is equivalent to 1 < (x/y) < 2. Thus we have found an equivalent fraction with smaller denominator, giving the desired contradiction. (x/y) = (Ny - kx) / (x - ky)
Fibonacci GCD's, please -- Math Fun Facts Fibonacci numbers exhibit striking patterns. Here's one that may not be so obvious, but is striking when you see it. Recall the Fibonacci numbers: n: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14,... fn: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377,... Now let's look at some of their greatest common divisors (gcd's): gcd(f10,f7) = gcd(55, 13) = 1 = f1 gcd(f6,f9) = gcd(8, 34) = 2 = f3 gcd(f6,f12) = gcd(8, 144) = 8 = f6 gcd(f7,f14) = gcd(13, 377) = 13 = f7 gcd(f10,f12) = gcd(55, 144) = 1 = f2 Do you see the pattern? gcd(fm,fn) = fgcd(m,n). Presentation Suggestions: After presenting the general result, go back to the examples to verify that it holds. The Math Behind the Fact: The proof is based on the following lemmas which are interesting in their own right. gcd(fm,fn) = gcd(fm,fqm+r) = gcd(fm,fqm+1fr+fqmfr-1) = gcd(fm,fqm+1fr) = gcd(fm,fr) gcd(fn,fm)=gcd(fm,fr) which looks a lot like the Euclidean algorithm but with f's on top! How to Cite this Page: Su, Francis E., et al.
Radice quadrata senza calcolatrice Il calcolo della radice quadrata senza calcolatrice è un procedimento algebrico che, mediante una serie di operazioni base, consente di calcolare l'approssimazione di una radice quadrata senza calcolatrice, o il valore esatto nel caso dei quadrati perfetti. Dopo aver visto, nella precedente lezione, cos'è la radice quadrata di un numero, spiegheremo in questo articolo come si calcola la radice quadrata a mano cioè senza l'aiuto della calcolatrice. Il calcolo delle radici senza calcolatrice non è facile e richiede esercizio e soprattutto pazienza, ma ci permetterà di calcolare la radice quadrata approssimata anche di numeri che non sono quadrati perfetti, cioè di quei numeri per i quali non esiste la radice qudrata esatta. Nella precedente lezione abbiamo visto che per calcolare la radice quadrata esatta di un quadrato perfetto basta ricorrere alla scomposizione in fattori primi. Mostreremo i passi da seguire calcolando la radice quadrata di 636.804. Scriveremo: Nell'esempio: 63-49=14.