Archimedes had been contracted by the tyrant King Heiro II to find a way to check if his crown was pure gold. The King had suspected that a bit of silver had been used in making the crown. Archimedes’ first solution, if the King was so sure, was to crush the crown into a small cube where the volume could easily be measured and compared with the mass. Density (the ratio of mass to volume) was a well known concept, as was the fact that gold is more dense than silver. The King would not allow this. Archimedes, in a huff, took to the bath where the famed moment occurred. Eureka! He had discovered that one can measure the volume of the object simply by measuring the volume of water that had been displaced.

This meant that he could take the crown and submerge it, then find the volume of the crown my measuring the displacement of water.

While the crown does not appear in the work of Archimedes, it is the most well known legend of the man.

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What spurs the advancement of civilization? This is easily a very complex question, however the simple answer we will use is *technology*. Civilizations advance as their technologies advance. Where once there were hunters and gatherers, tools and other knowledge allowed for the possibility of crop cultivation. This sort of technological innovation can be seen during every major change of civilization. At the heart of this innovation sits mathematics. But we couldn’t have gotten to civilization as we know it today just with basic geometry and counting.

So, what happened? Abstraction is an important key in the advancement of mathematics. Back to our students, as they progress through learning how numbers interact, they eventually reach the true first level of abstraction. Instead of learning how numbers interact with each other through examples, students learn how number interact with each other through ideas. It is at this point, our students have reached **algebra**. In algebra we focus on the ideas of how numbers interact with each other. We learn things like the distributive property, polynomials, graphing, solving equations, etc. These theorems that are developed and taught in algebra classes, show the students that numbers are predictable. Now instead treating each multiplication problem independently, they get to see that it is truly the same across the board. It is at this point our story splits and follows those students that delve further into the depths of mathematics.

Our math loving students will next make it in to **calculus**, where they have their first true dance with the real numbers. But the ideas of calculus are not restricted to just the real numbers as these students will find out. In fact, the real numbers are a part of a larger system (and our next level of abstraction) of **metric spaces. **Students begin learning the structure of metric spaces and how metric spaces interact with one another via maps. Often these ideas are imparted to the students in a mathematical analysis course. Many of the ideas of metric spaces are still not restricted to this system. In fact, we have reached the next level of abstraction, **topology**. Here, students are no longer concerned with numbers and distances at all. Instead they simply have a notion of “closeness” in terms of open sets. They begin working with highly abstract spaces, often those that can’t even be visualized, yet the ideas of topology allow us to assess the deeper structure of mathematics. For example, manifolds helps us identify types of spaces that would, on the small scale, “look” the same as the some product of the real numbers. This might say something about the shape of the universe itself. For example, we could be sitting in a manifold right now, but it “looks” like 3-space on our very tiny scale. But there are problems in topology that are quite hard to solve. Namely, determining which spaces are homemorphic (can “morph” into) to each other. But there are many tools in abstract algebra that, if phrased in the right way, will be able to help. There must be a way to marry these two fields.

This is exactly what Saunders Mac Lane and Samuel Eilenberg achieved with

*General Theory of Natural Equivalences*, Transactions of the American Mathematical Society Vol. 58, No. 2 (Sep., 1945), pp. 231-294 (JSTOR).

Though it was not widely accepted (and in fact rejected), this article was published as seen above. Mac Lane went on to publish Category Theory for the Working Mathematician in 1971, which is widely accepted as THE introductory book for the subject. As described in an earlier post Categories and Functors, category theory looks at the most basic relationships between mathematical spaces. Many common patterns emerge as proofs in one specific setting are seen applied quite easily to the universal setting. This in effect can prove many important theorems at once. But is this as far as abstraction takes us? Can we go farther? The answer is most certainly yes. See higher category theory. There may be no end to our abstraction as subject. One day, it might even lead us to the very nature of Mathematics itself, and thus the nature of the language of the universe. Who knows what we’ll discover!

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Many of us have seen the complex numbers. These are numbers of the form where and are real numbers and . Perhaps more important to this story, a complex number can be interpreted as points in a plane. Instead of the usual Cartesian plane with and axes, we have a Complex plane with a real and imaginary axis. So, for example, we can plot the complex number as the point .

This plane has many interesting geometrical concepts. But it is still just a plane. So yet again, we’ve hit a wall. What if we wanted a number system that works nicely, but applies to our 3-space? Enter William Hamilton.

William Hamilton was an Irish mathematician and physicist concerned with this exact problem. In 1843 Hamilton, likely in an attempt to clear his mind, went for a stroll with his wife. While walking on that October day, they decided to cross Broome Bridge. But little did Hamilton know, mathematics inspiration waits for no one. It suddenly dawned on him!

William, beside himself, was absolutely not prepared for writing down such a discovery. So, he did what any self-respecting mathematician would do. Gone was the care for public property as he carved hard these rules into the stone of the bridge, his face grinning gleefully as he did so. Unfortunately, no trace of the carving remains today. Instead, there is just a plaque commemorating the event.

The Quaternion, as he called them, found their use in mechanics and other areas of Physics. Multiplication in this group can be interpreted as a rotation in 4-dimensional space. For more on the algebraic side of things, consult popular Abstract algebra textbooks such as Dummit and Foote.

Hamilton also introduced the Bi-quaternion. This is an 8-tuple with a multiplication rule. One might be led to ask the question, is there a 16-tuple that works nicely? How high can we go? There are a ton of methods for extending the real numbers. Check out the Cayley-Dickson construction and go from there. However, if we are talking about finite dimensional division algebras, we can only go to the Quaternion and no farther. This is Frobenius’ theorem. Hyper-complex numbers is an ever interesting field of mathematics. These not only extend the real numbers, but also extend our understanding of the inherent nature of that we call number.

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If you happen to have a ribbon or something like it close by, lay it out flat. Twist one end 180° and connect it to the other end. Now look at the object created. The Möbius strip is two dimensional. Notice it only has a length and width. That means that it is a two dimensional object that exists in three dimensional space! Not only that, if you attempt to press your strip down on a table, it starts to cross itself and lose its properties. So the strip cannot exist in 2D space.

Imagine, or try, to color one side of this strip. You may have realized that if you tried to color just one side, you actually color the whole thing! This means that this surface has only one side. That is, the Möbius strip is non-orientable. To see this further, fix the strip still in space. Now take a point and give it a direction “outwards.” If you move this point around the strip, by the time you return to its original position, the direction will be reversed! See the applet from Math Insights for an interactive visualization. Not only is this strip non-orientable, it is in some sense a basis for determining non-orientability of other surfaces as noted in the following theorem.

**Theorem: **A surface is non-orientable if and only if it contains a subspace that is homeomorphic to a Möbius strip.

On this last note, we leave a short exercise. Draw a line in the center of your Möbius band. Cut along that line and explore what happens.

Topology is full of fun and interesting shapes and surfaces. In the next post, we will look at some other topological spaces you can make with a simple rectangle! We will be exploring the strange and the unusual in this exciting series!

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The anthem of Topological Data Analysis is that data has shape and that shape matters. We would like to take a data sample and describe the topological space it was sampled from. This will help us make predictions to where new data may land. But what tools do we need? As the name suggests we are going to be looking in our topology textbooks. We need an invariant of homeomorphisms that can be described easily by linear algebra. This is so because we wish to be able to actually compute these things.

Homeomorphisms would be ideal, however they won’t work. It can be quite difficult to even come up with a homeomorphism. So we look to the next best thing, homotopy. Again homotopy groups are difficult to even write down, we wouldn’t want to try to code them. Okay, well there is one more ‘H’ word that might help us out, and you may have guessed it! It’s homology. Homology is invariant under homeomorphisms, meaning if two topological spaces are homeomorphic, then they have the same homology. Hence, if two spaces do not have the same homology, then they cannot be topologically the same. Also, homology is easily describable through linear algebra. This makes it incredibly easy to compute! The main problem is that two problems with the same homology may still not be homeomorphic. This problem is handled with the principle of Occam’s Razor and will be explained later.

In this post we will give an introduction to simplicial complexes. We will also define two simplicial complexes, the Rips and Čech complexes, that are quite popular in practice. So let us begin with the definition of a simplex.

**DEFINITION: **An **m****-simplex ** is the convex hull of *m*+1 points (called **vertices**) in . We describe a simplex by its vertices. i.e. will denote an *m*-simplex.

From left to right we have a 0-simplex, 1-simplex, 2-simplex, and 3-simplex.

If is an *m**–*simplex, and is an *n*-simplex for *n* < *m*, and if the vertex set of is contained in the vertex set of we say . We are immediately able to build structures with these simplices.

**DEFINITION: **A **simplicial complex ** is a collection of simplices satisfying the following two rules.

- If and then .
- If then or .

The first property is commonly known as *downward closure*. We refer to the second one as a *minimal incidence* property.

On the left is a simplicial complex. The structure on the right fails property 2.

**DEFINITION: **A **simplicial map** between, is a map so that whenever is a simplex in , we have is a simplex in .

**LEMMA: ** **Simp**, with objects as simplicial complexes and morphisms as simplicial maps, forms a category.

*Proof: *The reader is encouraged to try to prove this using the definition of categories in Categories and Functors.

While it is great to have an idea for what simplicial complexes look like, this definition is actually slightly too concrete. For this reason, we need to define abstract simplicial complexes. The idea of these objects is that they are abstract enough to be useful in theory and computation, and it is okay to worry about the nice space that they will fit in later.

**DEFINITION: **An **abstract simplicial complex ** is a finite collection of sets satisfying, and implies .

Notice we dropped the minimal incidence property as well as the necessity of having a vertex set of points in Euclidean space. The notion of a simplicial map is exactly the same. Hence we obtain a category **AbSimp**. What we will see next is a justification for using abstract simplicial complexes. First note, that for every simplicial complex there is an abstract simplicial complex with the same vertex set of .

**LEMMA: **There is a functor so that for all and for all .

This lemma simply says that we can associate each abstract simplicial complex with a simplicial complex in a nice way. This allows us to work with these abstract objects and then fit them into a nice space later.

Now, we will take a look at two popular abstract simplicial complexes. One gives an accurate description of the space but is not easily computable, while the other is not as accurate but is easily computable.

**DEFINITION: **Let be a finite set of points in some metric space. Let be a positive real number. We define the **Čech complex at scale **to be the set of all simplices whose vertices lie in and the intersection of the balls centered at these vertices with radius is nonempty. In symbols

Where denotes the ball of radius centered at .

The Nerve Theorem tells us that our space is properly described by this complex. However, computationally, this complex is taxing. The problem is one is trying to find the intersection of metric balls which is much harder than just simply checking a condition. So we look to the next complex.

**DEFINITION:** Let be a finite set of points in some metric space. Let be a positive real number. We define the **Rips complex at scale **to be the set of all simplices whose vertices are within of each other. In symbols,

Notice the Rips condition is simply one we have to check. Though as previously stated it does not give an accurate description of the space since as we see in the picture we would fill the triangle in well befor the Čech complex would. However! It does it “well enough” as described by the following Lemma.

**LEMMA: (RIPS) ** .

Really this thing says that the Rips complex approximates the Čech complex well enough.

That will do it for part 1. It is meant to be an introduction to simplicial complex. Part 2 will cover enough Homology to understand Persistence. In the future we will look at concrete examples and the categorification of the field.

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The anthem of Topological Data Analysis (TDA) is that data has shape and that shape matters. We would like to take a data sample and describe the topological space it was sampled from. This will help us make predictions to where new data may land. TDA has been used in many fields such as medical imaging [1] , sensor networks [2], sports analysis [3], disease progression [4], image analysis [5], signal analysis [6], and many others.In this post, we are just going to give the basic idea. Suppose we have are given a data set that looks like this.

It seems obvious to the human eye that this data has been sampled from circular object. This is because we are wired to recognize patterns, especially ones as easy as this data set. But how could we get a computer to understand this pattern? This is where TDA comes in. Imagine that we begin growing balls around points.

As the balls grow they will intersect. When two balls intersect, we place a line segment (edge). When three balls intersect we place a triangle. When four balls intersect we place a tetrahedron and so on.

Eventually, the balls will have grown enough to bound a gap.

As we continue growing the balls, the gap will eventually close. Beyond this point nothing changes topologically, hence we can tell the computer to stop here. Now what we have done is created what is called a **filtration** which is simply an increasing chain of spaces. To capture the topological properties, we use homology to count holes. We apply homology (count the holes) to each space in the filtration. Then, more or less, we measure how long the holes last. The idea is that the longer lasting holes are more important to the topological properties of the space the data was sampled from. This process is accurately called **persistent homology**. There are, of course, some fine details excluded from this summary, especially the fact that TDA does not begin nor stop at persistent homology. If you would like to know more please check out some of the references I am leaving at the bottom. I will be making a post (or series of posts) soon that will go a little deeper in the theory.

REFERENCES

The first 6 references are applications of persistent homology.

[1] Lee, Hyekyoung, et al. “Persistent brain network homology from the perspective of dendrogram.” *Medical Imaging, IEEE Transactions on* 31.12 (2012): 2267-2277.

[2] De Silva, Vin, and Robert Ghrist. “Homological sensor networks.” *Notices of the American mathematical society* 54.1 (2007).

[3] Goldfarb, Daniel. “An Application of Topological Data Analysis to Hockey Analytics.” *arXiv preprint arXiv:1409.7635* (2014).

[4] Nicolau, Monica; Levine, Arnold J.; Carlsson, Gunnar (2011-04-26). “Topology based data analysis identifies a subgroup of breast cancers with a unique mutational profile and excellent survival”. Proceedings of the National Academy of Sciences 108 (17): 7265–7270.

[5] Bendich, P.; Edelsbrunner, H.; Kerber, M. (2010-11-01). “Computing Robustness and Persistence for Images”. *IEEE Transactions on Visualization and Computer Graphics* **16**(6): 1251–1260.

[6] Perea, Jose A.; Harer, John (2014-05-29). “Sliding Windows and Persistence: An Application of Topological Methods to Signal Analysis”. Foundations of Computational Mathematics 15 (3): 799–838.

The next few are just references for one who would like to get started in studying the subject.

[7] Edelsbrunner, Herbert, and John Harer. *Computational topology: an introduction*. American Mathematical Soc., 2010.

[8] Bubenik, Peter, and Jonathan A. Scott. “Categorification of persistent homology.” *Discrete & Computational Geometry* 51.3 (2014): 600-627.

[9] Lesnick, Michael. “The theory of the interleaving distance on multidimensional persistence modules.” *Foundations of Computational Mathematics* 15.3 (2015): 613-650.

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Our undefined terms will be **set** and the binary relation of **membership**, when we write it will read as “ is a member of . We will use capital letters and to represent sets.

**(1) AXIOM (of Extensionality) : **For every and for every and for every if implies and is implies then . This can also be stated as

.

We have here what it means for two sets to be equal. That is, two sets are equal if they contain the same members. The next axiom guarantees the existence of subsets in a way.

**(2) AXIOM (Schema of Comprehension): **Suppose is a property of z. For every there exists so that if and only if and is true. We see this as,

**DEFINITION: **The set is a subset of the set if and only if implies , in this case we write .

Suppose we are given . Then for any we are guaranteed, by the Axiom Schema of Comprehension, that there is some whose elements all satisfy . But since there is nothing that satisfies we conclude contains no elements. Notice, as of right now, our depends on a given . Suppose there is some other set, with no members. Then the statement is vacuously true as the antecedent is always false. We use the same reasoning to see that the statement . We conclude then by the Axiom of Extensionality that . (this is actually a proof of the proposition: “There exists a unique set with no elements”) Hence, the set with no elements exists and is unique, so we obtain the following definition.

**DEFINITION: **The set with no elements will be called the **empty set** and will be denoted, .

Notice the empty set is a subset of every set since the statement is vacuously true. The next axiom allows us to “build” sets in a way. It tells us that we can find a set containing two given sets.

**(3) AXIOM (of Pairing): **For any and , there exists so that and .

Notice that this axiom does not give us that contains only and . For this we have the following proposition.

**PROPOSITION: **For any pair there exists a unique set containing exactly and as its only members.

*Proof: *Let be a set containing and by the Axiom of Pairing. Let be the statement “ or “. By logic and the Axiom Schema of Comprehension we find that exists. We conclude that is unique by the Axiom of Extensionality.

From the Axiom of pairing we also get for any set the set is valid. And, we get that . This will come in handy later but first,

**(4) AXIOM (of Union): **For any set there exists a set whose elements contain all of those elements in the members of .

Let and be sets. We can use the Axiom of Pairing to obtain a set . Then the axiom of union guarantees exists. But what about intersections? We have actually had the definition of intersection all along. Let be the statement ““. Then we obtain by the Axiom Schema of Comprehension, there is a containing elements of for which is true. But this makes exactly the set of all for which and . Hence . We obtain the following definitions.

**DEFINITION: **Let and be sets. The **union** of and is the set where if and only if or $latex z\in y.

The **intersection** of and is the set for which if and only if and .

**(5) AXIOM (of Replacement): ** Let be a property such that for every there is a unique so that holds. Then for every there exists a such that for every there is a so that is true.

We can use this axiom along with the Axiom Schema of Comprehension to show that there is a set containing exactly the that satisfy . We can show it is unique by the Axiom of Extensionality. (Try this!)

**(6) AXIOM (of Infinity): **There is a set so that and whenever , .

From this we find , , and , and so on. Let be the property that is one of these sets. Then we define the set to be the set of all so that is true. This is no mistake for this set is exactly what we will call the natural numbers. In this way we replace these sets with notation, There are many more intricacies of this process (such as uniqueness) that we will not delve into in this post. For our subject is that of Zermelo-Fraenkel axioms. So we shall proceed from this topic.

**(7) AXIOM (of Power Set): **For every set there exists a set so that implies .

From this axiom (and the others we have used) we can define the unique power set of a set (prove it!).

**DEFINITION: **Let be a set. The **power set** of , , is the set which contains exactly all of the subsets of .

The next axiom will complete the list for Zermelo-Fraenkel (ZF) set theory.

**(8) AXIOM (of Foundation): **Every nonempty set (a set so that ) contains a member so that .

.

The Axiom of Foundation tells us that if is a set then . (Why?) Then we see that in this system Russell’s Paradox is not possible. The final axiom is the Axiom of Choice. This is a very controversial axiom, so in the interest of space we will simply state it in this post.

**(9) AXIOM (of Choice): **Suppose is a collection of nonempty sets. There exists a function that assigns to each set an element .

The axiomatic system formed by (1)-(9) is known as ZF+C or ZFC and is the most widely used and accepted system in mathematics today. In the next post we will explore the Axiom of Choice and its equivalents. We have seen that, by assuming the Axiom of Choice, it is possible to split a sphere into two spheres that are congruent to the original. The next axiomatic system we will look at is Bernays-Gödel-von Neumann axiomatics as presented in Dugundji’s* **Topology. *This will be the last set theory system we will explore for a while.

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It isn’t too often that we think about or even explore the fundamental building blocks of mathematics. These building blocks are called **axioms. **Axioms are statements taken to be true, i.e. they cannot be proven. This causes mathematicians, whether they know it or not, to take a lot of things on faith. All of modern mathematics is built on top of **axiomatic systems**, which is a list of axioms from which one can logically derive theorems. We further require that no axiom on the list can be proven by using other axioms. In this series we will look at some interesting axioms and axiomatic systems that govern mathematics. We will begin with perhaps one of the most important systems and that is sets. One could argue that the system of logic used is the most important. We are assuming a working knowledge of tools used in logic, and will forgo formalizing such a thing. We often think about an axiomatic system like a tree.

But first let us get a feel for why an axiomatic system is so important. This brings us to something called **naive set theory**. This naive set theory is an example of common occurrences in mathematics in which we define things by using natural language. We often wind up tossing words around until someone formalizes an idea. This happened with set theory.

**DEFINITION: **The collection of all things satisfying a property forms a **set**. In symbols, if is the statement “ has property “, then a set can be written in the form .

From this definition one can deduce that the collection of all things that are sets is a set. In symbols, if is the statement is a set, then is a set. But this means contains itself. Now that we know one exists, let us call all sets that contain themselves **abnormal**. In symbols, a set is abnormal if . Where means is an element of.

Obviously does not contain itself as a set. We will call these sets **normal**. In symbols, a set is normal if . It is clear that these two definitions are exclusionary and all sets are either normal or abnormal. Now consider the following set.

.

This is simply the collection of all normal sets. By definition, this is indeed a set. But now it must either be normal or abnormal. If is normal, then . By definition of we have which is a contradiction.

On the other hand if is abnormal then , hence by definition of we have $R\notin R$ another contradiction. We conclude that is neither normal nor abnormal. Which is our ultimate contradiction which tells us that naive set theory is inconsistent.

This example is known as Russell’s Paradox (1901) and it has many equivalents in naive set theory). Even though naive set theory has this major flaw, it is still very useful in teaching and introducing students to the real heart of mathematics. In the next post we will explore the Zermelo-Fraenkel (ZF) axiomatic system which is the most widely used system in mathematics today. We will also make note of ZFC which includes the historically controversial Axiom of Choice.

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Alternatively, it says that we can take a pea, break it up, and rearrange it to be the size of the sun. By now, one might ask themselves how something such as mathematics could support or even prove this outrageous claim! The answer lies with the Axiom of Choice, which we now state.

**AXIOM: **For every collection of sets there is a function so that for every .

The axiom simply says that we can assign to every set, an element of itself. This doesn’t seem that far-fetched, which is why it is generally accepted by mathematicians. In fact it has equivalent forms, such as Zorn’s Lemma, Well-Ordering Principle, and the Hausdorff Maximal Principle, which have been used to prove some of the most important theorems in mathematics. Such examples include:

- Every vector space has a (Hamel) basis.
- Every commutative ring with unit contains a maximal ideal
- The Cartesian product of of compact topological spaces forms a topological space with the product topology.
- Banach’s extension theorem which is used to prove the Hahn-Banach Theorem.

These are but a few of the consequences of the Axiom of Choice. One such consequence is the subject of this text. We will now formally state the Banach-Tarski Theorem.

**THEOREM: **Let and be bounded subsets of , for . Suppose further that the interiors of and are not empty. Then there exists some and partitions of and that can be written as and where is congruent to . In this case we write .

Put simply, if we have two bounded sets with nonempty interior in Euclidean space of three dimensions or greater, then we can break them up into sets so that their pieces are congruent. Now, there are some words that we should define just so we are all on the same page. We will go through those now. For the purposes of this post, .

A set is **bounded** if it is in some sense of finite size. More formally,

**DEFINITION: **A set in Euclidean space is **bounded** if , where is the Euclidean norm.

A **partition** of a set is a set of pieces which can be put together to make the set. Or,

**DEFINITION:** Given a set , a **partition** of is a finite collection of subsets so that

- ; and
- if .

The **interior** of a set is the set of all points that do not lie on the boundary (or edge). Formally,

**DEFINITION: **A point is **interior** **to a set** if there is some for which . The **interior** of is the collection of all such points.

Finally, two sets are **congruent** in Euclidean space if there is some way to fit one over the other without stretching, i.e. if they “look the same”. We define this formally as,

**DEFINITION: **Two sets are said to be **congruent** if there is an isometry (distance preserving, injective map) between them.

We will now prove the Banach-Tarski Theorem for a solid ball in in 4 steps.

**STEP 1: **Let be the (free) group of all strings containing 4 different symbols so that does not appear next to and no

appears next to a . In *vSauce’s *video, he calls these 4 elements . Now we will define a set to be the set of all allowable strings that begin with . We will define the sets and similarly. If denotes the identity then we can write as . This is because every string that is not the identity, must start with one of the four elements. Note that also the set is the set of all strings that do not start with . Hence we can write as . We can do this with the b’s as well. . So, what we’ve done here is taken the group , split it up into 4 pieces, then shift these pieces to create two identical copies of the original group. Now we have to find a structure in that is isomorphic to the .

**STEP 2: **We will consider our motions as rotations about an axis. To be more precise let be a rotation of about the -axis, and let be a rotation of about the -axis. There are other choices for but we shall content ourselves with this one. Now we let be the group generated by these rotations and . Note that this is almost the same as except we need to know that there are no nontrivial combinations of and that returns us to where we started. Let . Then it can be shown (see [2]), by using elementary linear algebra, that maps to where and . This means that . Hence what we have is that is isomorphic to .

It is at step 2 that we see why we must be in dimension 3 or higher as we cannot form this group in 2 dimensions. We will now start to see how the sphere fits in to this picture. This may get a little tough after this point.

**STEP 3****: **Let be the unit sphere with a partition generated by . Each set in the partition is called an **orbit**. We now apply the *axiom of choice *to choose exactly one point from each orbit. We collect all of these points in a set . Now we define four sets arising from .

- ;
- ;
- and;
- .

Now we have that where is the set of points that are fixed by rotations of . Similarly . Notice that any rotation has exactly two fixed points, and since is countable, there are only countably many points fixed by an element of , i.e. is countable. Now let be a line through the center of the ball that misses . Let be an angle so that , for all . Let . Then we see .

This tells us then that and .

We have made it out of the thick part of the woods now. We will now extend this result to the entire sphere.

**STEP 4: **We now extend this result to the solid sphere, minus its center simply by extending each point to a segment emanating from the center but not including the center (this is called radial extension). To finish, we consider the line obtained by shifting the -axis up by half the radius of the sphere. We next consider a rotation about with angle . It can be seen that whenever . Thus we collect all of these into a set . We conclude by the same reasoning as before that .

This is no trivial theorem by any means. Its consequence is indeed known as a paradox. However, note that we also relied on the sphere containing uncountably many points. It is not at all clear that there are uncountably many points in real life. But if we do, then it should be possible to duplicate objects. And if we don’t it would seem that Zeno’s Paradox should hold. It is interesting to think about the consequences of seemingly simple things. I encourage the interested reader to explore the references listed below.

[1] Wikipedia

[3] S. Wagon, The Banach-Tarski Paradox, Cambridge University Press, Cambridge (1985).

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Pythagoras of Samos (569-500 BCE) was an actual person, but was also the founder of the *Pythagoreans*. He was a political figure and a mystic. Beyond this, he stood out in his time as he involved women as equals in his activities. The Pythagorean society focused on mathematics, but also had some religious properties which include a rigorous list of tenets. Among these were,

- To abstain from beans
- Not to pick up what has fallen
- Not to touch the white cock
- Not to stir the fire with iron

and many more. Not only this but the Pythagoreans had great devotion to Number:

“Bless us, divine Number, thou who generatest gods and men.”

and

Number rules the universe.

The Pythagoreans also believed that each number held great meaning all numbers held a gender and meaning. Odd numbers were male, and even numbers were female. For example,

- The number
**one**: the number of reason. - The number
**two**: the first even or female number, the number of opinion. - The number
**three**: the first true male number, the number of harmony. - The number
**four**: the number of justice or retribution. - The number
**five**: marriage. - The number
**six**: creation - …
- The number
**ten**: the*tetractys*, the number of the universe

For more on the beliefs of the pythagoreans click this link. One may recall the name Pythagorean from a geometry course. In fact, it is one of the most well remembered rules learned in math. This is of course,

**Pythagorean Theorem: **In a right triangle, the sum of the square of the legs is equal to the square of the hypotenuse. That is, .

Evidence has been found that the Babylonians and Chinese both used this theorem 1000 years before Pythagoras’ time, however Pythagoras is thought to be the first to offer a proof of it. As previously stated, this is easily one of the most recognized theorems in the world. It has over 50 proofs. One of these by former U.S. President James Garfield.

Pythagoras noticed, however, that if he takes a triangle with two legs of lengths 1 he comes out with . He then wondered if there was a rational number to satisfy this identity. As the title suggests, his conclusion was

**Theorem: **There is no rational number whose square is 2.

*Proof: *We shall offer a proof of this theorem using modern mathematics. By way of contradiction, suppose there is indeed a rational number (where and have no common factors) so that , but this means,

which implies .

By definition the right side is even ( is even when is an integer) Hence the left side must also be even. Then we find that for some integer . Then,

which implies . Thus must also be even. But this is a contradiction to our hypothesis that and have no common factors as they are both divisible by 2. Thus we cannot have that is a rational number.

Q.E.D.

This revelation was very surprising to a society that worshiped Number. Everyone was wedded to the idea that numbers were rational, but here is irrefutable proof that there are numbers that are irrational (it was found much later by Cantor that there are in fact “way more” irrational numbers than rational numbers). For a while, the Pythagoreans treated this as a secret. As the legend goes, Hippasus of Metapontum was murdered for revealing this secret. A nearby community started to become wary of the growing popularity of the Pythagoreans. So much so that war was waged on them. It is thought that Pythagoras perished in the fire of his own school. Numbers in the past were important, and they are just as important today, but is it possible that a mathematical idea can still be so outrageous that it could spark a war?

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