My story of surfaces starts in a beautifully weird morning when I got up to realize that life in the usual Euclidean plane had changed dramatically. Vectors had shortened, areas had shrunk, and infinity was just a few feet away!

Here is the aftermath (why the term “after math?!”):

What had happened was this: A vector of unit length in the usual sense was now much shorter, of length where is the distance (in usual sense) of its base (=initial point) from the origin. Yes, here it does matter where the base of a vector is.

The vectors that used to be perpendicular are still perpendicular. By looking at parallelograms, we see that areas have shrunk by a factor of . So, in the above diagram, the square feet of the two shaded areas are in fact almost equal.

Let me do some measurements.

**Length of an infinite straight ray.**

Consider (see picture.) To calculate lengths we must add up lengths of tangent vectors. Tangent vectors are all , unit length in ordinary plane at each point, so their new length must be taken into account. The result is:

Hmm! Seems that the infinity is only feet away!

**The area of the whole plane!** (But wait, isn’t that infinite?)

We add up areas of infinitesimal rectangles . As discussed above, in our new *metric *(Ok, I had to finally use the term!) this rectangle’s area is shrunk by our factor. So, we get

Maybe my whole success in posting these series depends on my ability to convince you now to experiment with this new plane I have created. Be bold about it. Ask questions, do calculations for yourself, get a feel of the structure of this space. Ask what could be asked about this space? What other familiar constructions are possible on it? Could we talk about spaces of functions? What is the measure? How will we integrate a function? What is the shortest path, say between and ? Notice that going on the straight line isn’t the best, because we are better off bending away from the origin so that lengths are shorter. But how much to bend?!

And finally answer this: the plane with this tampered metric is a copy (in almost all aspects) of a familiar 2-d shape, can you guess what shape?!

To be continued with more fun…

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This month’s riddle is actually fairly well-known, but was communicated to me by C. Cook. As always, if you have a nice mathematical riddle that you would like to share, then let us know and we’ll see if we can post it!

You, an intrepid explorer, have found a mystical fortune-telling shrine. This shrine is particularly mystical because it has *three* mystical oracles (all in the same room), making it manifestly superior to other run-of-the-mill mystical shrines, which have only one oracle. Complicating matters further, however, is the fact that not all of the oracles in the shrine tell the truth. **In fact, exactly one oracle always tells the truth, exactly one oracle always lies, and exactly one oracle always answers randomly.**

Your job is to determine, using no more than three true-or-false questions, which oracle is which. Each question must be addressed to a single oracle, and may (if you wish) be about the other oracles in the room. You are required to ask only questions which have a definite true-or-false answer, even if you suspect the oracle to whom you are addressing the question is the random oracle. (Note that a question such as “would Oracle A say ‘true’ to such-and-such a question” would thus be allowed only if you were certain that Oracle A was not the random oracle.)

It is not required that each of the three questions be asked to a different oracle (so you could ask the same oracle all three questions). Successive questions may of course be based on the answers to previous questions.

**Can you come up with a set of three questions that will allow you to determine the identity of the oracles? **Give your answers in the comments below!

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What does Arrow’s Impossibility Theorem actually say? Like Euclid’s parallel postulate, Arrow’s Theorem is frequently transformed into a variety equivalent formulations that are either more intuitive to grasp or more relevant to problem at hand (to read Kenneth Arrow’s original paper, click here). The formulation below is based on a paper by John Geanakopolos, who according to the acknowledgments, consulted Kenneth Arrow prior to publication.

Consider a society of N voters and at least 3 candidates. Each voter has transitive preferences, meaning they can mentally rank all the candidates (with ties allowed). A voting system is a system that produces a single ranking on behalf of society (i.e. it determines a winner, but also a second place runner-up, third place, etc.). Consider the following three fairness criteria:

**Universal Domain:**Voters can rank their candidates in any order they like. No matter what combination of ballots it receives, the system will always produce a ranking.**Unanimity:**If everyone prefers A to B, the voting system should rank A above B.**Independence of Irrelevant Alternatives (IIA):**Whether the voting ranks A above B should not depend on how voters rank C (or any other candidate).

**Theorem:** **The only voting system that respects universal domain, unanimity, and IIA is a dictatorship.**

Note: The dictator in this case is not a candidate, but a rather voter who has the power to decide the winner every time, no matter what the rest of society thinks (more precisely, a dictator can always force the voting system to rank A above B).

A two-page proof, accessible to any outsider who is used to proving things, can be found here. To give you a taste, I’ll walk you through the first lemma (or you can skip to the next section see why Arrow’s Theorem might not be so doomsaying after all):

**Lemma:** **If every voter ranks one candidate as their highest or lowest choice, then the voting system must as well.**

*Proof:* Suppose, on the contrary, that even though every voter has put B either first or last, the voting system ranks A>B>C. Thus every individual voter ranks B>(A and C) or (A and C) > B. Suppose every voter decides they like C better than A, but leaves B’s rank unchanged (they’re allowed to rank them however they want under *universal domain*). By the *independence of irrelevant alternatives* condition, the voting system must still rank A>B and B>C. Now, every voter’s mental ranking is now C>A>B or B>C>A. By the *unanimity *criterion, the voting system must rank C > A. The two rankings A>C and C>A are not compatible so our original assumption is impossible. The voting system must rank B either first or last. QED

The proof then uses this lemma to show that there is a unique voter who can cause B to go from being last to first in the rankings and that this voter must in fact be a dictator.

Before we throw up our hands and declare that no voting system can be fair, let’s pause and consider how reasonable these assumptions and “fairness” criteria actually are. Unanimity does not sound like an assumption we ought to relax: directly contradicting the wishes of literally every voter sounds about as bad dictatorship. Likewise, we could limit the voting system’s universal domain, but prohibiting voters from ranking candidates a certain way because it will cause the system to malfunction hardly seems fair either. If we force there to only be two candidates, then all our contradictions go away, but presumably you will need some system of primary elections to narrow your choices down to two, so this only kicks the can back up the road. Even if we don’t need our voting system to produce a ranking but rather just a single winner, unanimity affects affects “winner” just as much as any other choice. But what about IIA?

On its surface, this criterion seems like the sort of fairness we would want in a system, though like Euclid’s parallel postulate, it does seem a bit more complex than the other basic assumptions. Most systems around the world violate IIA. The 2000 Gore versus Bush election, for example, likely constituted an IIA violation: that is, presence of third party candidate Ralph Nader tipped the election from Gore to Bush, even though the Nader voters did not necessarily like Bush more Gore. This does seem unfair, at least to Gore and Bush, though not necessarily for the voters. Arrow’s Impossibility Theorem assumes the people actually vote according to their mental ranking of the candidates: that is, they vote *naively, *not *strategically. *If you assume these Nader voters knew the potential outcome of their actions (and that’s a big “if”), then the system was taking into account their preferences: that is, they cared so much about Nader that they were willing risk Gore losing even if they preferred him over Bush. Note that Arrow’s Impossibility Theorem only takes into account *rankings*, not *ratings**. *Allowing voters to rank each candidate on scale of 1-10 (or grade them A-F), a system known as range voting actually avoids Arrow’s paradox altogether. However, if at least some of our voters are strategic, then this system is highly manipulable (e.g. even if don’t think your second choice candidate is so bad, you give her an F if you think the election is going to be close). In fact, the even more consequential (though less famous) impossibility theorem known as **Gibbard–Satterthwaite Theorem**** **states that every voting system is either:

- A dictatorship,
- Prevents one of the candidates from ever winning, or
- Is vulnerable to strategic voting (i.e. savvy voters can tip the election by misrepresenting their true preferences).

Again, whether strategic voting is a bad thing depends on your point of view. It does seem unfair in that some voters who are less informed might naively vote their true preferences, while others may vote strategically and exert an undue influence. Even if everyone had perfect information, some voters (say, voters who prefer a hopelessly small party) are in a much better position to swing the election to their second choice than anyone else. But perhaps this is a difference we can live with—after all, a voter who is fiercely dedicated to one candidate and will vote for them come hell or high water is far less influential than a swing voter who can be persuaded to change their vote, no matter what the system. Still, it seems prudent to try to limit the advantage that a small group of savvy, strategically-minded voters have over the rest of us, just as it seems prudent to lessen the chances of an irrelevant alternative causing an upset.

This observation then, bring us to our solution, if you can call it that, to the seeming impossibility of a fair system. While in theory no system can meet all the fairness criteria we may desire, our challenge is to design a system that will minimize the likelihood of an unfair outcome. Thus, our “perfect design” problem becomes an optimization problem, and luckily there are many candidates: instant runoff voting, approval voting, and the Condorcet method to name a few (Wikipedia’s coverage of these topics is extensive). The plurality system (the most common one used in the US) is almost certainly *not* the optimal solution since its highly manipulable by strategic voters and violates a number of other fairness criteria, though its simplicity does have great appeal. Instant runoff voting, used in Australia and some local U.S. elections, has a number of weird flukes, such as a vote for your favorite candidate actually hurting their chances, though it’s unclear if this is more of a concocted scenario that is likely never to pop up in the real world. **“Most systems are not going to work badly all of the time,” **Kenneth Arrow stated, in the lead up to the 2008 election.** “All I proved is that all can work badly at times.”** (See here for full discussion this paragraph is based on). Perhaps one of you will find a system that is likely to never behave badly in real world conditions. While that may sound modest, finding such a system could still be a huge contribution of mathematics to the the fairness of democracies around the world.

Works Cited:

Arrow, Kenneth J. “A difficulty in the concept of social welfare.” *The Journal of Political Economy* (1950): 328-346.

Geanakoplos, John. “Three brief proofs of Arrow’s impossibility theorem.” *Economic Theory* 26.1 (2005): 211-215.

McKenna, Phil. “Vote of no confidence.” *New Scientist* 198.2651 (2008): 30-33.

Graphics in public domain, available: https://en.wikipedia.org/wiki/Voting#/media/File:Vote_12345.jpg

Also consulted: https://en.wikipedia.org/wiki/Arrow%27s_impossibility_theorem

]]>One immediate benefit of considering coordinate-free descriptions of geometric objects is that we may talk about “curves” that are not *a priori* embedded in . In other words, we don’t have to start with a subset of to be able to study 1-dimensional objects. There is already quite a nontrivial question we can ask: what curves can be embedded in a plane? The answer will be provided as a condition on and , and this description has the advantage of having nothing to do with or any other non-intrinsic data. Later we will talk about surfaces (2-dimensional manifolds) that do not live in 3-space, but rather in 4-space. Having an intrinsic way of seeing objects is liberating and opens up new possibilities.

**A 1-dimensional manifold**

It may also happen that the data are equivalent only on subsets of the intervals. For example, it could be the case that the data restricted to the first half of interval is equivalent to the data on the last third of interval . Then in this situation, we may “glue” together these overlapping compatible parts and get a longer curve that extends the one on . There will be no ambiguity in measurements over our new curve due to the equivalence of two sets of data in the intersecting parts. This possibility of patching together pieces while maintaining the structures is a fundamental part of the concept of a manifold.

Our definition will be of a Riemannian 1-manifold (because of the metric structures we have decided to keep). A Riemannian 1-manifold is a Hausdorff topological space such that each point in has an open neighborhood homeomorphic to an interval in , along with the set of data as above. If two open neighborhoods have a nonempty intersection, then we require that on the intersection the two localizations be equivalent in the sense of the previous section. There is also the condition of *second countability*: We like for our manifold to be covered by countably many of such neighborhoods.

**Coming up…**

Later in this series on manifolds, I will turn to surfaces which are 2-dimensional objects and re-interpret much of their calculus in the intrinsic language of differential geometry.

And if you missed them, here are Part One and Part Two of this series.

Let us know if you have any questions on manifolds in the comments below!

]]>After a particularly grueling example involving a word problem and a piecewise function, I asked my students, “What questions do you have regarding the process that we just went through?” and gave them a couple of minutes to review their notes and formulate questions. Gripped by the mid-semester complacency, my students lethargically looked through their notes and shook their heads indicating to me that there were no questions. Instead of taking their word and moving on, I was inspired to not move on and instead ask a question that would lead to discussion and had the potential to pull them out of the complacency that they were experiencing.

The first test was still recent enough that my students remembered the feeling of not knowing how to complete a problem on it so I drew from that experience. I looked out across the sea of students and told them what their lack of questions meant to me was that they were all going to get a problem like this correct on the next exam.

Suddenly my lifeless class had life, and throughout the room there was laughing and many audible no’s. I latched onto the no’s and asked again where were their questions. Immediately I got feedback from the students about where they were struggling and we had an excellent class discussion regarding the questions they had and how to overcome those struggles. Since then, the number of questions in class has increased and the students overall seem more engaged with the material.

Obviously this is not a fix-all solution and I wouldn’t recommend using it all the time or even frequently because it will just become routine for the students. They will expect it and just nod their heads that they can do it. However after a particularly difficult example it might be something to try if you are struggling with student engagement and complacency like I was. After all it is a new question and they can’t just answer like they usually do, and you can catch them off guard and elicit a whole new level of engagement from students that wasn’t there before.

What strategies have you used in your classroom to get your students to genuinely engage? Share your tips in the comments below!

]]>It seems like all anybody can talk about right now is the election. And while it has definitely given me a lot to think about in terms of political, cultural, and social problems in America, there’s also some interesting and potentially troubling math behind our electoral system that I think deserves attention as well. I want to explore election math just a bit, demonstrating that structural changes can seriously affect what kind of candidate wins, and questioning how elections should really be organized.

The Electoral College (https://www.archives.gov/federal-register/electoral-college/about.html) was originally established as a compromise between a full democracy, which would just count the popular vote, and having members of congress elect the president. In most states, the winner takes all of the electoral votes, although a few (Minnesota, Nebraska) assign votes proportionally. The process, as I’m sure you know, begins with primaries, and concludes with a race between the winner of each primary and any third party candidates.

Two years ago, I attended an excellent talk at the Joint Math Meetings in San Antonio, where Prof. Donald Saari discussed many different forms elections can take. The overall point is that, depending on the structure of an election, many different candidates could win the election with the exact same voter preferences. For example, consider last year’s Republican primaries. For simplicity, let’s just look at Trump, Rubio, and Bush. Suppose out of 100 voters, voter preferences followed this pattern:

40 voters: Trump, Rubio, Bush

25 voters: Rubio, Bush, Trump

35 voters: Bush, Rubio, Trump

To be clear, I’m making up these numbers to illustrate a point that I think is prescient.

Now, in a plurality rules primary, Trump wins with 40% of the vote, despite 60% of people, a clear majority, having him ranked last!

Consider another election system, where people rank the candidates, giving their first choice 2 points, their second 1, and their last 0. Assuming nobody tricks the system (and I’ll acknowledge that this might be a faulty assumption, but let’s go with the thought experiment), Trump gets 40 points, Bush gets 95, and Rubio gets 125. This gives Rubio the win, with a pretty large margin.

Consider yet another setup, where Rubio and Bush are grouped in a party, and Trump is in his own. Suppose Trump’s voters vote in a separate primary just electing Trump, and Bush and Rubio compete with the 60 voters who rank Trump last. Bush will win this primary by 10 votes. Then in a general election between Trump and Bush, Trump gets 40 votes and Bush gets 60, so Bush wins.

These are three setups- plurality, rankings, and primaries. Each one has a different candidate winning, but which one makes the most sense? Which one best captures voter preferences? I’m not arguing that we should immediately change the system, because I genuinely am not sure which I think is most mathematically or politically justified. My point is just that, especially after such a volatile election cycle, perhaps our democratic methods deserve greater scrutiny. Our government is deeply divided into parties that cannot seem to work together, and I think more moderate leadership, or at least leadership committed to working across party lines, would be productive and incredibly positive for America.

]]>In the previous segment, we concluded with the fact that a curve in may be viewed as an interval *I* together with the following set of data:

- A real-valued function , which will help measure the length,
- A real-valued function , which will help measure the curvature, and,
- A real-valued function , which will help measure the torsion.

In this segment we will see some examples and discuss possible constructions that this allows us.

**Examples:**

The data represents the part of the parabola between . More precisely, this segment of the parabola is the only curve with the above curvature and torsion; any other curve is just the same one with a change in the position of the observer in .

The data represents the helix

**Equivalence of Curves**

Often, the first thing one does after defining/constructing a new object in math is to define an appropriate notion of “sameness” between two of them. To this end, we have the following definition:

*Definition***:** Two curves and are equivalent if there exists a diffeomorphism such that

In calculus language, this would be a “re-parameterization” of a curve. If this happens, then the two curves that they describe coincide completely — they will overlap after a rotation and translation.

The above definition is derived from the plausible requirement for the following to hold:

A change of variable in the second integral leads to the first condition in the definition above.

**Example**:

We saw from our previous example that describes a part of the parabola .

It turns out that the set of data also describes the same curve, with an explicit equivalence being given by the map .

**Intrinsic Integration On Curves**

Suppose that we are given a function which assigns to each point on a curve a real number, say the temperature at that point. We wish to find the average temperature over the curve. To do this, we need to add up (integrate) the temperatures at each point and divide the result by total number of points (length).

The following is an appropriate definition of the integral of a real-valued continuous function on our manifold:

Again, it is adding up values of on , just as in Riemann sum, except that now we have a non-uniform weight applied to each summand.

The reason this definition works is that if we calculate the integral using a different but equivalent set of data, then by the change-of-variables formula, we would get the same answer:

*Observation: *Our definition of integration only depended on the function . Therefore, it suffices to have a way of measuring distances in order to be able to define a parameterization-free, -free notion of integration on a curve.

*Note: *The integration above may be familiar from calculus, or complex functions, as a “line integral”. But it is usually not emphasized there that this is intrinsic to the curve. However, it indeed comes with the curve, independent of the embedding of the curve in

**What’s next …**

Now that we know a way of telling when two parameterizations coincide, in the next installment, we will be able to “glue together” pieces to arrive at a manifold.

The definition of the integration will work globally because different parameterizations agree on it locally.

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Always making me think: Why is this still a thing?

Okay, certainly there are legitimate reasons to collect demographic information, including information regarding gender. Whether to explore the diversity of the field (i.e. graduate admissions or university surveys) or whether it is needed for grant purposes (i.e. conferences) there are times when inquiring about an individual’s gender identity is reasonable. However, why must this information be collected in such biased fashion?

The fact of the matter is that for many individuals gender is a complicated social construct, with which they have a nuanced relationship. As phrased above, the question “Gender?” takes all of these complexities and nuances and *smashes* them; *smashes* them into two extremely narrow little circles. It takes individuals and *smashes *their agency, forcing them to squeeze themselves into a narrow space predefined by someone else.

Moreover, to some who are “gender-expansive”, this question is yet another instance of someone — a colleague, a peer, a fellow mathematician — neither recognizing nor accepting part of who they are. It is yet another instance in which they are in part invisible:

“invisibility may seem like a small price to pay … But invisibility is a dangerous and painful condition. When those who have the power to name and to socially construct reality choose not to see you or hear you … when someone with the authority of a teacher, say, describes the world and you are not in it, there is a moment of psychic disequilibrium, as if you looked into a mirror and saw nothing. Yet you know you exist and others like you, that this is all a game with mirrors.”~Adrienne Rich, “Invisibility in Academe” [1]

In short, this question invalidates the feelings and experiences of those who do not find themselves comfortably within the confines of the two narrow little bubbles provided to them.

If I agree there are legitimate reasons to collect gender demographics, and if I find the above format highly problematic, what do I think the answer is? Well I think a better approach is to give agency to the respondents. As the Human Rights Campaign notes the least restrictive and most preferable option is to allow individuals to self-identify. For example, by re-phrasing the question in the introduction as follows:

Doing this empowers the respondent to define their gender identity as they see fit. I should also point out – not that it should matter – that phrasing the question in this way changes little how the question writer interacts with the responses. (Note other parts of academia have already begun moving to eliminate such biased and discriminatory questions.)

Some of you might be wonder why I have bothered spending ~500 words on a topic so “insignificant” as to how to ask someone their gender. The fact of the matter is that these things are not “insignificant” – they matter, and they have substantial effects on the lives and careers of many people. (I am reminded, in part, of the following tweet.) Sure, this might not have an effect on your life, but that does not mean the opening question does not make others feel uncomfortable, outcast, and discriminated against. And to say this is “insignificant” is to say these people are, well, insignificant …

Which is unacceptable. No person is insignificant. No one should ever be made to feel insignificant. No one should ever be … invisible.

For us to create a more inclusive, supportive, and welcoming mathematical community, these are the sorts of things we need to begin to think and care about. We need to stop seeing things like this as “insignificant”. We need to begin to see the world and individuals as they are: complexly. Most importantly, we need to step up and change the ways we act. Changing how we phrase this question is a small first step.

[1] Roberts, T. et al. *The Broadview Anthology of Expository Prose 2 ^{nd} Ed. *Broadview Press. 2011.

There are competing theories online about possible interpretations of John von Neumann’s quote, but *manifolds *are definitely some mathematics that “you don’t understand … you just get used to them,” — at least for a while.

In a series of posts reflecting on my own experience, I will try to motivate the conceptualization of manifolds, and the implications such an abstraction has/had on our understanding of, basically, shapes. I hope to point to some beautiful geometry in low dimensions that you may have passed by too quickly to take notice of.

I must underline the subjective nature of my articles, and that by no means are they meant to narrate the history of the subject, or depict a current fashion in the community. This is simply “another article.”

The first three articles will be dedicated to converting the conventional calculus of curves into manifold language. We will see that a curve can be replaced by an interval endowed with some structure. This will pave the way for an exposition of the theory of surfaces in subsequent articles. The reason for such an extended sequence is to include as much detail and as many examples as possible.

** The Question**

**What We Know from Calculus**

From calculus we fix a Cartesian coordinate system for three-space and then parameterize our curve by a map from a subset of ,

If is a smooth parameterization, then the length of the object between two of its points and is given by

The curvature is given by

and the torsion is given by

where differentiation is taken coordinate-wise.

Notice that the integral above involves *only the* *length *of . Thus, even if we don’t have itself , but only some , we will again be able to measure the length of any segment of our curve.

This observation invites a search for a representation of our curves independent of the three-space.

**The Answer to the Question**

In a search for a 1-dimensional embodiment of a curve, the following theorem is the best we could hope for.

**Theorem:** Given differentiable real-valued functions , and , with ranging in an interval , there is a curve with being its curvature and being its torsion and giving its length between points. **Moreover**, any other curve with this description is obtained by changing our origin and rotating the axes rigidly. (See Chapter 1 of do Carmo’s book *Differential Geometry of Curves and Surfaces.*)

Thanks to this theorem, an interval along with the following set of data can be thought of as a curve in :

- A real-valued function , which will help measure the length,
- A real-valued function , which will help measure the curvature, and,
- A real-valued function , which will help measure the torsion.

Note that this data is independent of the specific positioning of our curve in 3-space.

**So far…**

A curve in is nothing more than an interval in along with three real-valued functions defined on it! Once we have this set of data, we can forget about our original curve as a subset of , and work in 1-dimension. After all, we can reconstruct an exact copy of our curve in 3-space from the data whenever we wish.

**In the Next Installment…**

- We will discuss some constructions, e.g integration on curves, that this description makes possible;
- We will see examples of curves described by a set of data,
- We will find out a way to tell when two sets of data determine the same curve, which will allow for the definition of 1-manifolds in the following article.

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If you’re interested in writing some of your own mathematically-oriented posts, be sure to let us know by contacting someone on the editorial board directly or using the Contact page on our website!

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