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Engineers, mathematicians, and physicists all love the Fourier transform. When we hear about the Transformers movie, we keep thinking about Fourier Transformers, with Oscilloscope Prime leading the noble Autocorrelationbots against the nefarious Decepticonvolutions. Look heroic (and nerdy) with our Oscilloscope Prime math shirt!

The Fourier transform converts functions of time into their frequency components. In other words, it expresses the function as a superposition of a continuum of complex waves of pure frequencies.

If you clap your hands near a piano, how will this excite the strings? You can imagine sustaining a single note on a flute near a piano, and this might resonate with just the string at that frequency while leaving most of the other strings of the piano largely unaffected. The Fourier transform says that the clap can be viewed as a superposition of many flutes playing different pitches at different amplitudes. If we understand how the piano reacts to each flute, then we can understand the reaction to the clap.

Convolution is a natural but complicated operation. The convolution of two probability distributions is the distribution of the sum of independent random variables with those distributions. When you multiply two numbers, you convolve the digits, and then carry. If the digits are small enough, then there are no carries and you only have to convolve the digit string. When you multiply 1011 by 203, you get the convolution 205233. The Fourier transform of the convolution of f and g is the pointwise product of the convolutions of f and g. While convolution is a complicated operation on functions, it is simple on their Fourier transforms.

Noise such as static in an audio recording or scratches on a photograph may affect primarily the higher frequencies. To improve the signal-to-noise ratio, one approach is to dampen the higher frequencies relative to the lower ones. To do this, we can convolve with a function g whose Fourier transform is 1 on low frequencies, and 0 on high frequencies. Since the Fourier transform applied twice returns the original function multiplied by -1, we can find this useful g by taking the Fourier transform of a function which is 1 when -1< t < 1, and 0 outside this interval. The Fourier transform of this function is the function sinc 2x, so convolving with a sinc function is one way to reduce noise.

Our picture of Oscilloscope Prime shows a rectangular wave displayed on his oscilloscope form, and its Fourier transform, a sinc function displayed on his chest. This math shirt can be purchased via our nerdy shirt site.

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If there is a 1-1 correspondence between a set and the integers, then the set is called countable. Cantor proved that the real numbers are not countable, while the rational numbers are countable. The integers are obviously a subset of the natural numbers, but there is a way to label the rationals with integers so that each label is used exactly once. These results are covered in a college introduction to real analysis, but without giving an explicit construction. This is a shame, since the Calkin-Wilf tree produces a simple, natural bijection between the positive integers and the positive rationals.

To construct the Calkin-Wilf tree, start with 1/1. This is a binary tree, and every node has two children. The left child of 1/1 is 1/2, and the right child is 2/1. The left child of a/b is a/(a+b), and the right child is (a+b)/b. The descendants of 1/2 are 1/3 and 3/2. The descendants of 2/1 are 2/3 and 3/1.

Euclid’s algorithm is a classical method for finding the greatest common divisor of two numbers. Any number which is a factor of both 10 and 24 is also a factor of their difference, 14, so GCD(10,24) = GCD(10,14) = GCD(10,4) = GCD(6,4) = GCD(2,4) = GCD (2,2) = 2.

The Calkin-Wilf tree is related to Euclid’s algorithm, since the descendants of a/b are the fractions whose (numerator,denominator) pair leads to (a,b) under Euclid’s algorithm: (a,a+b) and (a+b,b). Since every positive rational number has a unique form p/q with p and q relatively prime, and Euclid’s algorithm reduces (p,q) to (1,1), every positive rational is contained in the Calkin-Wilf tree precisely once.

If we read the rows of the Calkin-Wilf tree in order, 1/1, 1/2, 2/1, 1/3, 3/2, … we get a list containing each positive rational number exactly once. Where does p/q appear? To find the binary expansion for the location, perform Euclid’s algorithm on (p,q) to reduce it to (1,1), and record a 0 when you replace (a,a+b) by (a,b), and a 1 when you replace (a+b,b) by (a,b). Add a terminal 1, and then reverse the digits. This is the binary expansion of the location n in the list.

The denominator of a/(a+b) is the same as the numerator of (a+b)/b. Surprisingly, this pattern also holds between adjacent fractions of the list which are not descendants of the same fraction, e.g., the right child of 1/3 is 4/3, and the left child of 3/2 is 3/5, and the denominator of 4/3 is the numerator of 3/5. (We highlighted this by using the same color.) This means the list of numerators 1,1,2,1,3,… determines the list of fractions, as the nth fraction is (numerator n)/(numerator n+1). What is this sequence of numerators?

The nth numerator is the number of ways of expressing n-1 as a sum of powers of two, where each power of two is used at most twice. For example, 6 can be expressed in 3 ways, as 4+2, 4+1+1, and 2+2+1+1. This provides a combinatorial way to construct the Calkin-Wilf tree, and to produce a bijection between the natural numbers and the positive rationals.

We chose the Calkin-Wilf tree as the subject for one of our math shirts because of its connections to multiple areas of mathematics, and because it deserves to be better known.

This math shirt can be purchased via our nerdy shirt site.

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The Fundamental Theorem of Algebra states that any nonconstant polynomial has a root in the complex plane. In fact, a degree n polynomial has n roots, if you count roots with multiplicity, but it suffices to show that there is at least one root. The Fundamental Theorem of Algebra was proved by Gauss in his doctoral dissertation.

To prove this using topology, we need the idea of the winding number of an oriented curve in the plane about the origin. This is a precursor to the fundamental group of a space. The winding number of a curve which does not pass through the origin is an integer which counts how often the curve goes around the origin in the clockwise direction. The winding number does not change under continuous deformations through curves which do not pass through the origin.

If the constant term of the polynomial is 0, then 0 is a root of the polynomial, and we do not need to prove anything more. So, let use assume the constant term of the polynomial is c != 0.

Consider the image of circles about the origin under the polynomial. Small circles will be mapped into a small neighborhood of c, and will thus not wind around the origin at all. The images of small circles have winding number 0. Sufficiently large circles will behave like the largest term, and can be deformed to their image under z|->zⁿ, where n is the degree of the polynomial. That winds n times around the origin.

As we increase the radius from 0 to infinity, the image changes continuously, and the winding number changes from 0 to n. Since the winding number is preserved under deformations which do not pass through the origin, and n>0, there must be some radius r so that the image of the circle of radius r does pass through the origin, so there must be some z with |z|=r so that the polynomial sends z to 0, a root. QED

This math shirt can be purchased via our nerdy shirt site. Many colors and styles available!

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At first glance, the shirt on the left looks like a soccer jock shirt – but it’s actually a very nerdy math shirt.

A soccer ball (European football) usually has 12 pentagons and 20 hexagons, arranged in a highly symmetric fashion. The polyhedron most closely resembling this familiar object is the truncated icosahedron, an Archimedean solid.

Archimedean solids are an extension of the Platonic (regular) solids. Archimedean solids have regular polygons as faces, and have symmetries which send any vertex to any other. The rotational symmetries of the truncated icosahedron are the same as those for an icosahedron or dodecahedron. There are 60 rotational symmetries, isomorphic to the alternating group A5.

The truncated icosahedron is also the shape of buckminsterfullerene, C60, the buckyball. This is a form of carbon with fascinating properties only identified in the 1980s. The discovery of fullerenes earned the 1996 Nobel Prize in chemistry. Some viruses also share this shape.

On our Truncated Icosahedron Math Shirt, we have chosen to show the Euler characteristic equation χ = vertices – edges + faces = 2. The surface of any convex polyhedron is a sphere, hence will have the Euler characteristic of a sphere, which is 2 for a 3-D polyhedron. For the truncated icosahedron, this equation is 60 – 90 + 32 = 2.

The Euler characteristic is a deep topological property connected with the Lefschetz fixed point theorem. An application to the sphere is that any orientation-preserving map from the sphere to itself must have at least two fixed points, counting multiplicity. As an example, if you rotate the sphere, there are two fixed points where the axis of rotation intersects the sphere. Another example: The complex plane plus infinity is also a sphere. The map z |-> z + 1 has only one fixed point, infinity, but this fixed point has multiplicity 2.

The Euler characteristic χ can be computed from any decomposition into polyhedra, so it is a constant for convex polyhedra. For a cube, vertices – edges + faces = 8 – 12 + 6 = 2. For a dodecahedron, vertices – edges + faces = 20 – 30 + 12 = 2. That the Euler characteristic is constant is easily guessed, but harder to prove. Elementary proofs don’t explain the importance of the Euler characteristic.

One advanced explanation for why the Euler characteristic is constant is that it is the alternating sum of the ranks of the homology groups of a space. Any map deformable to the identity, or whose action is the identity on homology, will have at least χ fixed points, counting multiplicity.

The Gauss-Bonnet theorem says the total curvature of a surface equals 2π times its Euler characteristic. A smooth sphere of radius 1 has unit curvature per area, so its total curvature is its surface area, 4π, or 2π * the Euler characteristic 2. A discrete version covers the truncated icosahedron. At each vertex, the sum of the angles of the polygons is less than 2π radians. Any polyhedral decomposition of a surface has total angular defect equal to the 2π times the Euler characteristic. If 3 polygons meet at each vertex, then the angular defect may be associated to the polygons instead of the vertices. The defect of an n-gon is 2π * (1-(n/6)), so the defect of a hexagon is 0, the defect of a pentagon is π/3, and the defect of a quadrilateral is 2π/3. Thus, both the dodecahedron and truncated icosahedron have 12 pentagons, to have total defect 4π, and both the cube and truncated octahedron have 6 squares.

Another symmetric sports ball is from sepak takraw, a popular Asian sport which may be described as kick-volleyball. The classic design for the takraw (ball) has snub-icosahedral symmetry, or icosahedral symmetry without the reflections. The hollow (whiffle-like) structure is made from reeds or plastic pieces woven around 12 pentagons in a chiral fashion, so that takraws are either right-handed or left-handed.

The Truncated Icosahedron math shirt is available for purchase via our nerdy shirt site.