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Charles Darwin pointed out many vestigial parts of the human body in The Descent of Man. Vestigial does not mean that the parts have no conceivable use, but rather that they are vestiges or remnants of a more functional past organ.

The coccyx was one of Darwin’s original examples. The coccyx is the “tailbone,” an extension of the spine beyond the pelvis. In fetal development, the coccyx starts out as separate bones, but as we age it fuses into a single inflexible unit, sometimes fused to the sacrum above.

The coccyx is a vestigial tail. Its development is regulated by the same genes which regulate the development of tails in primates with tails. Its development and shape is similar to that of animals with small tails such as hamsters. There is a muscle, the extensor coccygis (see the classic anatomy reference, Gray’s Anatomy), which connects the top of the coccyx to the bottom, which would flex the tail if it were not fused.

It does not make sense to view the coccyx as an intelligent design. While the coccyx has some uses, such as the attachment point for some muscles and a weight-bearing structure when we sit, it could serve without closely resembling a tail, or starting as separate bones and then fusing. Having a muscle “designed” to flex a fused, immobile bone makes no sense.

By itself, the coccyx is not a logical proof that evolution occured. It is merely evidence which makes sense within the theory of evolution, and which makes no sense in an intelligent design. Human anatomy as well as the natural world is filled with such evidence. The only way to end up with a theory of intelligent design is to start with it, and then blind yourself to the contrary evidence. That’s not science.

The evolution shirt pictured above shows the rear view of the human pelvis, coccyx highlighted, with the slogan “Creationists can kiss my coccyx!” This evolution shirt can be purchased via our nerdy shirt site.

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To us, nerdiness means appreciation and understanding of science, mathematics, and technology. We nerds appreciate the beauty of art and nature, but also the elegance of powerful ideas. We think the world becomes more wondrous and more interesting as we understand more of it.

Sometimes nerdiness is expressed professionally, but you don’t have to be a scientist to enjoy science, as shown by the strong popularity of the Discovery Channel, NOVA, and other programs aimed at the interested public. We’re big fans of some areas of science we didn’t study in school.

There is a negative connotation to nerdiness in society, which associates weaknesses to every strength. Attractive people are assumed to be stupid. Strong people are assumed to be clumsy. Intellectuals must be antisocial and weak. These stereotypes are sometimes right, and sometimes wrong. We celebrate the positive aspects of nerdiness, and reject the negative connotations as wishful thinking of the insecure.

In some parts of the English-speaking world, nerd carries the good meanings, and geek carries the bad. In other parts, the reverse is true. To us, a geek is more specialized while a nerd is more general, and both are positive.

We’ve designed a special Nerd Pride shirt featuring the (visible) electromagnetic spectrum. The electromagnetic spectrum is a familiar sight from physics texts from elementary school through graduate school, since light is the most important tool we have for investigating the world. Coincidentally, rainbow flags have been used by numerous movements (indigenous peoples, cooperatives, peace/anti-nuclear, gay/lesbian/bi/transgender, etc.), often to symbolize pride in diversity or hope. As wonderful as we find Weird Al Yankovic’s “White and Nerdy” song, nerds are found in all colors, in all walks of life, and with all educational backgrounds.

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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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Reconciling the quantum mechanical behavior of subatomic particles with the ordinary behavior of macroscopic systems is still difficult, and this is the point of the thought experiment known as Schrodinger’s cat.

In a purely Newtonian world, particles would have a location and a velocity. If you send particles through a pair of slits, each particle either goes through one or the other. A standard college physics experiment shows that electrons do not behave like this. The electrons act as though each electron passes through both slits, illustrating the wave-like nature of electrons. At the moment the electron passes through, you can’t specify the location as a point. The location is a superposition of states.

While it is much more difficult to imagine macroscopic systems which behave as superpositions of quantum states, quantum mechanics predicts that this happens. Schrodinger’s cat is a hypothetical cat in a black box whose fate depends on the state of a subatomic particle. The state of the cat is a superposition of the living and dead states until you look at the particle or cat.

In a Newtonian world, probabilities would be real densities. In a quantum mechanical world, probabilities are described by complex distributions, the squares of whose magnitudes total 1. This allows destructive interference, as seen in the double-slit experiment. That’s why an even mixture of ψ_alive and ψ_dead is (ψ_alive + ψ_dead) / √2, not (ψ_alive + ψ_dead) / 2.

We mixed a depiction of Schrodinger’s cat with the internet LOLcat meme, which combines adorable images of cats with captions in broken English, usually as spoken by the cat. In the LOLcat vernacular, the wavefunction of the cat is: ψ_{IM IN UR BOX} = (ψ_HAI + ψ_{OH NOES!}) / √2. We have a non-LOLed Schrodinger’s cat shirt as well.

Modern transistors are based on quantum mechanical effects. This allows the basic switches to be extremely fast and small, but our computers are still Turing machines. They still follow the same model of computation as the old vacuum-tube giants, or pencil and paper computations.  Future computers called quantum computers may rely on manipulation of particles in superpositions of quantum states. These are fundamentally more powerful than Turing machines, and they may allow polynomial-time solutions to problems which take exponentially long on Turing machines.

The Schrodinger’s LOLcat and Schrodinger’s Cat shirts can be purchased via our nerdy shirt site.

To celebrate object oriented programming, we designed a shirt for infants and expecting mothers. The syntax is from C#, a popular object oriented language.

person baby = new person();

This line creates an instance named “baby” of type person, and calls the constructor for a person. A constructor is a special function which returns a new instance of a class. The colors match those used by the Visual Studio Integrated Development Environment for highlighting key words like “new” and class names like “person.”

baby.love();

This calls a method (function associated to the object) of the baby called “love.” It could be an action of the baby, or it could be something you do to the baby. We’ll leave the details to you.

The geek onesie and matching maternity shirt can be purchased at our nerdy shirt store on CafePress.

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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.