## What I wish the NCTM journals were

It’s hard for me to separate what I like doing as a writer from what I enjoy reading. Back when I was critiquing the NCTM journals it felt like I was being sort of vague about what I wanted…well, I had to be vague, because what I want to read are roughly the kinds of things that I write. (Emphasis on kinds; I don’t want to read me, I want to read you.)

But just to put the pieces together, here is what I would love to read in an NCTM (or any other) math education journal:

Toss in a few features — crosswords, math problems, a curriculum resource, etc. — and this is the sort of thing that I, Michael Pershan, would be interested in reading. I have no idea who else would read or pay money for this sort of thing, so please NCTM don’t rashly decide to change the journals to more closely resemble my blogging output. But this is close to what I imagine would be my ideal publication, i.e. the sort of pretentious garbage that’s my calling card.

## Reading “A Root Cause of the Teacher-Diversity Problem”

Check it out here, it’s great.

So far the main effect of having a second kid in the house has been to make me, in pretty much every way, a little bit dumber. You know what I mean by dumber: not as clever, etc.

Anyway, one of the ways in which I’m dumber is that I’m a worse reader with worse impulse control. So last night I found myself reading an article right before bed (another mistake) and having trouble understanding a perfectly clear article by Melinda D. Anderson (the one I mentioned). And then I had the questionable decision to tweet the author my questions.

I really feel for journalists on Twitter. How do you know when to engage, when someone is worth talking to? Melinda was very generous in helping me understand what she wrote, as was Grace Chen, so I figured I’d write a bit explaining what I understood, out of thanks.

***

About 15% of public school students are Black:

About 16% of public school students were Black in 2012, but only 7% of teachers were Black.

(Figures and sources from this NCES report.)

So even as Black people make up about 12% of the general population in the US and even as that share of the population is modestly rising, the percentage of Black teachers is not increasing. A disproportionate percentage of the teaching population is not Black.

The question is, why? There are roughly two sorts of answers.

Supply-side: The issue is in the pipeline. This is mostly what people talk about, and it takes myriad forms: Black people graduate high school and college at lower rates, and hence are less available for teaching positions. Black people are disproportionately uninterested in becoming teachers, for whatever reasons.

Indeed, that NCES report that I linked to focuses on the pipeline:

But there’s another set of factors to look at: what if the teachers are there, but schools just won’t hire Black teachers?

Demand-side: Districts are biased, either implicitly or explicitly, against Black teachers. For reasons rooted in stereotype or false-associations, Black candidates are less appealing to schools, so they don’t get hired.

That’s what Melinda wrote about in the Atlantic piece. She wrote about districts that were sued for being biased against Black teachers, and the court agreed. (Or, rather, one district was issued a decision by a court to stop discriminating against Black teachers, and another took action after they got spooked by a suit that was in the works.)

One thing I was wondering is whether it’s really possible to disentangle supply and demand considerations. If Black teachers experience bias in school hirings then they’ll be less likely to pursue careers in schooling. If you start with a dearth of Black teachers in the pipeline, biases will creep up. These factors might not be possible to truly separate.

My own school has disproportionately few non-White teachers (and students). Is the issue supply-side or demand-side? Both, probably, though I suppose it only matters to the extent that this helps a school decide what to do. As Grace points out, most in education prefer to frame this as a “won’t somebody think about the children” issue, i.e. it helps kids to have Black teachers. I’m not saying I’m skeptical of that, but as Grace notes this could be a way for schools to avoid responsibility for bias — the dearth of Black teachers is then not unlike having not enough computers or too few pencils.

But how do you tell a school to stop being biased against Black job candidates? The answer for both of these schools was affirmative-action policies. Is there a stronger case for affirmative action in the presence of bias than if the problem is in the pipeline? I think so, though I feel fuzzy about how to think about this.

Many people I talk to who are otherwise strong advocates for diversity in the teaching force are uncomfortable with affirmative action. But maybe there’s an important difference to folks between affirmative action to correct a present bias vs. affirmative action to address a historical bias?

I’m confused. Melinda said at one point in our Twitter chat that she wrote the piece to provoke thinking and discussion, and it’s definitely done that for me.

## Trying to Understand the Second Paragraph of “A Concise Course in Algebraic Topology”

In my attempt to understand the first paragraph and the notion of a topology (and a topological space) I realized that there was something I wasn’t getting. I understood the definition of “open” subsets of X, I also understood the meaning of “neighborhoods,” and I felt like I was getting a grasp on the axioms.

But what does it mean to have a “notion of nearness”?

Now, I think I’m starting to get it. In Calculus or Real Analysis you define a continuous function in terms of distance and the “I can get closer” game. But the point is that you don’t need distance in order to define continuity or a lot of those other analytic notions. All you need is a “notion of nearness,” and if open sets and neighborhoods make sense in your space, then you can usefully obtain that notion of nearness.

The usual Calculus definition of a continuous mapping goes like this: if we want all the values in the range to be within a certain neighborhood — no matter how narrow the neighborhood — we can always find a suitable neighborhood in the domain whose points all map to that band of points in the range. In this, a neighborhood is defined in terms of distance — wanting to be in a certain neighborhood in the range is being a certain (potentially tiny, epsilon) distance away from a point in the range.

But who needs to define a neighborhood in this way? We can replace this metric notion of neighborhood with a topological one, the one that we struggled to grasp in the previous post, that really comes down to open subsets. So, in short, a function is continuous at a point in a topological space if, no matter the neighborhood in the range (call it Y), there’s always a neighborhood in the domain (X) that the function maps to a neighborhood that’s a subset of the original (Y).

I should probably be trying to make these images myself rather than stealing them from wikipedia but whatever:

So now I’m thinking of topological spaces as metric spaces, minus distance. Which explains something I never really understood, which is why in topology you can deform all sorts of things into other things. Why should a coffee mug and a doughnut be equivalent in a topological sense? The answer is that distance doesn’t exist in a topological space. All we have is open sets and subsets of open sets which stands in for our notions of nearness. As long as a mapping (like a homeomorphism) preserves the open set structure — the idea that things that start out near each other should remain near each other, relatively speaking — then there is a topological sameness that is preserved.

Honestly I should probably just be learning topology instead of trying to make sense of this algebraic topology text, but let’s see how far we can go. The next paragraph makes the jump to algebraic topology.

## Trying to Understand the First Paragraph of “A Concise Course in Algebraic Topology”

I.

Wikipedia is here, and quite helpful in expanding on this definition. Here is my attempt to rewrite it in my own words, fleshing it out in ways that are helpful and natural to me.

What is a topological space?

Imagine that you have some collection of points that are in the Cartesian plane. Maybe it’s a circle that’s centered around the origin. Maybe it’s a line, like y = 2x + 3. Maybe it’s a grid of points falling perfectly along the chassis of the plane.

In a way, we can also imagine the city that you live in or the state or country as a collection of points. My apartment is one of these points in space, and I’m part of a neighborhood — Washington Heights. But there are other ways of talking about where I am. I’m not just in Washington Heights, but in NYC, in NYS, in USA, on Earth. There are points, arguably, south of where I live that are a part of two neighborhoods — Washington Heights and Harlem. These neighborhoods aren’t coextensive, but they do have some overlap.

Maybe there are some folks living out in the sticks that are part of no neighborhood. They’re off the grid, so to speak.

Let’s imagine a computer program that knows how to assign every point in a map of NYC to the neighborhoods to which it belongs. Let’s call that program N. You give N a point, and it gives you a list of neighborhoods.

Imagine if we let N loose on any of those spaces that we were talking about before — the disk, the grid of integer points, the portion of a line — and then let’s say that any of those spaces is a topological space if when N is applied to the space, nothing weird happens. Here is a checklist of four weird things that could happen:

• You apply the computer program N, but there’s a place that gets assigned neighborhoods that it’s not in. Like, maybe the program assigns a spot in the middle of the Hudson River to Washington Heights, even though a random spot in the river isn’t really a part of our neighborhood.
• Imagine that N says that my apartment is in Washington Heights, but not in NYC. That’s messed up. Bad N. If you can imagine a place like that, it’s not a topological space.
• If I lived a bit further north, I might be a part of two neighborhoods: Washington Heights and Inwood. N should assign me not just to Washington Heights and to Inwood, but to a new, hybrid neighborhood: Inwoodton Heights. If N can’t really invent such a neighborhood, then I don’t really live in a topological space. This one starts moving away from my regular intuitions about neighborhoods, clearly…
• …and so does this last test. As I keep on mentioning, my apartment is in a neighborhood Washington Heights. If N assigns my apartment to WaHi, then it should also report that there’s another neighborhood that I’m a part of that’s more local than WaHi, like my block. My block has the property that (a) I’m in it, it’s my neighborhood and also (b) everywhere on my block is also part of Washington Heights. If there isn’t such a place, I don’t live in a topological space.

Moving away from apartments and cities, let’s think about the x-axis. Real estate agents who are trying to get people with \$ to move to a place decide what neighborhoods mean in the real world. But what does a neighborhood mean on the x-axis? Let’s define a ‘hood of a point (like 3.1) to be any open interval (like between 3 and 5) that includes the point.

Let’s make sure that our computer program N would work alright in such a place:

• If you’re a neighborhood of a point, that point is in it. Yay!
• If you’re in an interval (like between 3 and 5) and that interval is in a bigger interval (like between 1 and 6) then you’re also in the bigger neighborhood.
• If you’re part of two neighborhoods (like between 3 and 5 and also between 3.05 and 7) then you’re also part of the overlapping neighborhood (between 3.05 and 5).
• If you’re part of a neighborhood (like between 3 and 5) there is a smaller, more local neighborhood that you’re part of (like between 3.05 and 4.95) and, more to the point, there always will be.

Huzzah — it’s a topological space!
II.

One more twist: this way of thinking about topological spaces is not standard any longer. People prefer to talk not about neighborhoods but instead about open sets. So we have to make sense of that, even though it’s been nearly a decade since we took Real Analysis.

Another wise wikipedia page says, “an open set is an abstract concept generalizing the idea of an open interval in the real line,” which makes sense. It tells you the delimit of a set, but it doesn’t include its boundary.

This isn’t enough, though. Since we defined a topological space in terms of neighborhoods, we want a definition of “open set” that relates “open set” to neighborhood. We need to connect these two concepts.

[Here’s where I got a little bit lost, so I went looking for help from another source. Google google, ended up at the Math Stackexchange.]

It doesn’t take much to merge the two concepts, as far as I can tell from what I’m reading. To fit the rules of neighborhood assignment, your potential neighborhood has to pass the following test: if a point is assigned a particular ‘hood, the point actually has to be in that ‘hood. (Fancy talk: if N is in N(x), then x has to be in N.)

An open set adds one slight additional requirement: if you’re in N, then N’s your neighborhood. All this eliminates is the possibility that a neighborhood is “too big,” including not just Washington Heights but also a random stretch of the Hudson River where nobody lives.

This is a definition that will do the trick for us, and allow us to connect the old-fashioned (but apparently useful for beginners like myself) definition of topological space to the new and trickier (but apparently useful for topology pros) definition that is couched in terms of open sets.

Here’s the open set version: still imagine a map of some space (like NYC) and still imagine that the map contains a bunch of points (like my apartment and other peoples’ apartments) and still imagine that there are ways of grouping those points that come with the map (like neighborhoods)…

…or don’t, and instead imagine some section of the x-axis and the Cartesian plane, and imagine that there are ways of making subsets of that section of the line or plane…

And now there are three rules about the neighborhoods (or subsets):

• The empty set is in it — i.e. the neighborhood of nobody is a neighborhood
• You can’t escape the space via the union of subsets — i.e. mush together any of the neighborhoods (subsets) and you’re still in a neighborhood (subset)
• You can’t escape the space via the finite intersection of subsets — i.e. be in as many of the neighborhoods (subsets) as you wish to at once and you’re still in a neighborhood (subset)

Not entirely obvious to me yet why these two definitions of topological space are equivalent. I see that they both have the intersection, I see how the intersection of disjoint sets would imply that the empty set is in the topology (but what’s wrong if there’s only one set in the topology according to the neighborhood definition?), and I guess the superset/subset axioms must cover the “closed under union” axiom of the open set definition? I’m going to let that slide for now.

III.

What are some examples and non-examples of a topological space? We already mentioned the x-axis, equipped with open intervals.

In fact, for things like the x-axis — metric spaces — the textbook page already tells us that we can think of topological spaces as things where the neighborhoods are like open intervals. They’re open disks, or open spheres, that act precisely in the way you’d expect them to. No funny business, no weird stuff.

As the textbook also says, we should think of this all as an attempt to point at what it takes to have a space that captures our feelings about “nearness.” So if something’s near, there’s always something nearer, that seems to be the most important part to me.

But whenever there’s a new idea, we need some examples and contrasting non-examples to set our heads on straight, and this is no exception.

The x-axis — i.e. the real line — is an infinite collection of points, but there’s no reason why a topological space needs to be infinite. And if our map of points is finite, there’s no reason why our computer program — our function — N has to be anything but a list of neighborhoods.

So, suppose you have four points in your space: 1, 2, 3 . What neighborhood assignments would result in this being a topological space, and which wouldn’t? Here is a helpful image from wikipedia:

• If you care your space into two sets — {} and {1, 2, 3} — then you haven’t really done much carving, but that’s a topological space, the trivial topology. That sort of captures the idea of nearness in an absolute way — everything is near, nothing is not, like a terrible party. (No idea if that analogy makes mathematical sense, by the way.)

Thinking about the function N and the first set of axioms: the same and only neighborhood has been assigned to the three points and they’re part of it, there are no supersets or subsets, so trivially we’re done.

• In contrast, if you carve your space into {}, {1, 2, 3}, {2} and {3} (bottom left) that’s not a topological space. From the point of view of the second definition, the issue is that the union of {2} and {3} isn’t included, so it’s not a topology. From the point of view of the first, the problem is that {2, 3} is a superset of {2}, and every subset containing a neighborhood must itself be a neighborhood. (Is this right?? It doesn’t feel right.)
• What if your topology contains {}, {1}, {2}, {1, 2}, {1, 2, 3} (middle left)? The second definition is indeed sort of easier to use for these discrete examples. It’s pretty clear: the union of {1}, {2} is {1, 2} and that’s in.

I need more examples, and wikipedia has a few more:

• Let the space be all the integers, and let the collection of subsets be all the finite subsets of the integers, any list of them, plus the set of all the integers. So, for example, if you have a given point like -5, there would be all these open intervals that we’d want to say are the neighborhoods of -5, like “integers between -10 and 2” and “integers between 0 and -10000,” and we’d also say “all the integers.” That wouldn’t work, though, because you the union of a bunch of these subsets isn’t necessarily part of the topology. Take the union of all the finite sets that don’t contain zero — {-2, -1, 1, 2}, {1, 5, 6}, etc. — that’s an infinite set, but there’s only one infinite set in the collection (all the integers) and this can’t be that infinite set. Therefore, it can’t be in the collection, and this can’t be a topology.

The tricky thing for me is relating this all to the talk that a topological space preserves our notion of “nearness.” Is it possible to relate each of these axiom failures to a failure of our notion of nearness? Personally speaking, my notion of nearness no longer operates when our space has just four points. Is there a good way to think about this that I’m missing?

TO BE CONTINUED, WHEN I TRY TO READ THE SECOND AND THIRD PARAGRAPHS OF THIS TEXT AS I ATTEMPT TO UNDERSTAND A PROOF OF BROUWER’S FIXED POINT THEOREM. STAY TUNED

## Little Shoes, Never Worn

For sale:
baby shoes,
never worn.

Machine washable,
slight stain,
probably pee.

Baby fine;
shoes not.

baby has
huge feet.

Truly big,
flippers really.

small hands,
small nose,
giant feet.

Never worn
baby shoes
for sale,
or just take them.

## My Answers to “Six Questions Math Educators Need to Answer Honestly”

[Questions can be found here.]

1. What is mathematics?

Don’t know. Pass.

2. Where should we learn mathematics?

Don’t know what “should” means here. Pass.

3. Should mathematics be a compulsory subject?

Question: What would the immediate effect of changing of making math non-compulsory tomorrow would be?

We’ve had decades (centuries?) to learn a bit about how to teach what we teach. I’m not saying it’s easy to teach systems of inequalities well; I’m saying that if I’m going to teach systems of inequalities there are dozens of resources I can turn to, curricula that have thought carefully about how to put the pieces of the course together to reach many students.

Have you ever taught a new course or an elective? It’s hard, isn’t it, not knowing how to go about the work. A world where we radically change the curriculum or where we eliminate mandatory classes is that world. It’s not like number theory and graph theory is easier to teach than polynomial division. Yeah, number theory is really cool, but it’s hard to turn cool things into learning. That’s the whole art of teaching.

So go ahead, make math non-compulsory. Honestly, I don’t care, what we ask kids to study in school is mostly arbitrary anyway. In the 1920s it was unclear whether math would be a compulsory subject in high schools — NCTM was created, in part, to protect math compulsory math offerings. If they’d lost that fight, who knows what hoops we’d be asking students to jump through before graduating. Mandatory home ec? Mandatory workshop? What reason do we have to think that these mandatory offerings would be better taught?

And if we make most of the curriculum non-mandatory? That’s like college, and if you don’t like how k-12 is taught then, wow, get excited for higher ed.

The fact is that there are tremendous pedagogical benefits from having a lot of people teaching and learning the same thing. School requirements are arbitrary, but there are benefits to keeping things the way they are.

4. How should we learn mathematics?

5. Who should learn mathematics?

Don’t know what “should” means here, pass.

6. What is the purpose of mathematics education?

Tempted to pass, but I think there’s something interesting to think about.

The purpose of math education depends on the purpose of math, and the purpose of math has varied across history, across cultures, across math departments, varied so much in the time of Fermat that there wasn’t even a single thing called “mathematics.” From The Mathematical Career of Pierre de Fermat:

With due allowance for exception, one may sort those practitioners into six broad categories: the classical geometers, the cossist algebraists, the applied mathematicians, the mystics, the artists and artisans, and the analysts. Although only one or two of these categories constitutes what one might call a “school” of mathematics, and although the work of many individuals falls into several categories, nonetheless each category distinguishes itself from the others by characteristic attitudes toward the nature and purpose of mathematics, its problems and methods of solution. Each category has a distinctive style, and the different styles often conflict in essential ways.

When people talk about the “values of math” or the “purpose of math education,” know that the question is ill-formed. Math education emerges from the competing needs of students, parents, communities, our government. It’s inherently contradictory, and given that the conflicts are genuine it probably should be sort of a mess. Teaching is not hampered by these conflicts, teaching is the job of making sense of these conflicting needs. That’s the job.

Math in 2018 is used for joy, pleasure, getting rich, pursuing justice, attacking foreign governments, passing tests, teaching, and dozens of other purposes too. When educators say “we have to get back to the values of math” that’s a lateral move; that’s no clearer. Math has as many purposes as math education.

“Math education is a very young child of mathematics,” Singh writes, a completely incorrect statement. One thing I learned from Learning Modern Algebra is that pedagogy is a major source of mathematical innovation. Part of why you’d need a way to find Pythagorean triples is to help you come up with nice problems for your students.

Singh calls math education “a spoiled and rotten child” of mathematics, as opposed to mathematics, whose purposes are clear and pure:

Mathematics is not up for debate. It is what it is, and it has been tattooed in so many civilizations and cultures for many millennia. Its purpose and mandates, might run parallel to math education at times, but in reality, they operate on a higher plane of truth, justice, beauty, play and love

Historically speaking, this is nonsense. Math changes, math has different purposes. Math is contradictory, math education is contradictory. Math is beautiful and ugly, so is math education. In math education we’re not the spoiled child of math — we’re the grown sibling, sometimes in touch, sometimes not.

So I’m entirely against Singh’s idea that we can look to the values of math and measure math education against them. In sum: Stop pooping on math education and educators. The end.

## The Bible According to My Son, Who Likes Both The Exodus and Dora the Explorer

CHAPTER 3

1     When Pharoah learned what Moses had done he rose to kill him, but Moses fled to the desert.

2     The LORD sought Moses in the desert, but could not find him.

3     He could not find him wherever he looked. The LORD looked over here, and also over there.

4     And the LORD said, “I can’t find Moses no matter where I’m looking.” So he went to ask Dora.

5     Dora spoke unto the LORD, “I know how to find Moses. Check the map.”

6     And the MAP spoke unto the LORD saying, “If there’s a place you got to go, I’m the one you need to know,”

7     “I’m the map. I’m the map. I’m the map.”

8     The LORD thanked Dora very nicely and then he found Moses. “There you are,” He said.

9     “You’ve got to go back to Pharaoh, Moses. Otherwise you’re going to be in big trouble. I’m going to get mad at you, I’m going to yell and scream at you. And you have to tell Pharaoh: ‘Pharaoh, you’re a bad guy.’ Tell him that.”

10     And then the LORD went with Dora and they had arroz con leche with abuela and they sang songs and had fun.

## In Love with Being Lost

Sunil Singh writes:

What is mathematics? If I would have answered this question before I began my teaching career, I would have been provided some cliched, stock answer revolving around its importance to many disciplines like science, engineering, economics, etc. — basically it being some practical workhorse. Now? Well, since I wrote a book about it, it is simply about happiness. Finding sporadic moments of bliss learning something new and wonderful about mathematics. Just playing with numbers, puzzles, brain teasers, conundrums, algebraic ideas, etc — mucking around in the sandbox of math. The more I know, the less I know. The less I know, paradoxically, the better I feel about my ideas about mathematics.

There’s no doubt that mathematics involves happy moments. But does Andrew Wiles’ description of mathematics sound like happiness?

Perhaps I could best describe my experience of doing mathematics in terms of entering a dark mansion. You go into the first room and it’s dark, completely dark. You stumble around, bumping into the furniture. Gradually, you learn where each piece of furniture is. And finally, after six months or so, you find the light switch and turn it on. Suddenly, it’s all illuminated and you can see exactly where you were. Then you enter the next dark room…

Not that this is necessarily inconsistent with joy. Maybe you get your kicks wandering around anonymous dark mansions — there’s something for everybody. But you’ve got to admit, it sure doesn’t scream out FUN.

So which is it: is mathematics all about happiness or about stubborn frustration? Well, why not both? If, as Singh suggests, we’d best think of mathematics as an “art” then we get to ask, what other art is only about happiness?

I think educators sometimes emphasize feelings of joy and experiences of beauty in math at the expense of all the other things you can feel or see. For a while, I’ve been interested in visualizations of ugliness in math — is such a thing possible? And, as the Wiles quote shows, mathematics often involves far more than happiness.

“Of course math involves more than happiness,” you say, “but the point is that happiness is the goal, the purpose, the carrot that mathematicians are always chasing.”

Here’s an idea though: what if what we’re really in love with is the feeling of being lost in a dark mansion?

## Stopping by Ft Tryon Park, Way the Heck Uptown, on a Snowy Afternoon

So you’ve got woods? Yeah, we’ve got those.
Well, not exactly, but something close.
A park with hills and rocks and trees
That people flock to whene’er it snows.

Your woods are dark, deep, and lovely?
I think our park you’ll find quite comfy.
Your little horse would love it here
I’ve heard the woods get quite lonely.

He’ll ask if he can stay and play
Romp with kids and ride their sleigh.
When we get cold we can go home
To watch snow fill our alleyway.

Woods do sound nice – save your pity,
The park’s also very pretty,
And no one’s lonely in New York City.
And no one’s lonely in New York City.

## Generalizing Circles and Ellipses

[Reading and thinking about: n-Ellipses and the Minimum Distance Sum Problem, a paper that I was able to stick pretty well with until about the last third.]

## 1. Generalizing Circle Definitions

There often comes a point in my geometry classes where I ask students to come up with a definition that describes a circle. It’s always interesting to see the ideas they land on. Some students try to capture the perfect curviness of a circle; others talk about the circle’s symmetry. When I share the “textbook” definition (which kids rarely come up with) there are sometimes sighs of relief or yelps of excitement in the room:

Circle: Take all the points that are a certain distance away from a center point — that’s a circle.

It’s a beautifully simple definition, and it’s one that leaves plenty of room for generalization.

If we’re looking to generalize the circle, a great next step is to analogously define an ellipse. I like to imagine the center of a circle splitting in half and turning into the two foci of the ellipse.

(Sorry for the glitchy gif.)

We can define an ellipse in a way that is analogous to the “textbook” definition of the circle:

Ellipse: Take all the set of points that are a certain distance away from two focal points — that’s an ellipse.

Just to clarify, what I mean by “certain distance away” is we’re looking for the sum of the distances to the two focal points. So if our “certain distance” is 10, then a point that is 3 away from one focal point and 7 away from the other would be part of the ellipse.

## 2. Generalizing Circle Constructions

Sometimes you see geometry problems about goats on a rope. The point being that the rope-stuck goat is restricted by a circle. That goat could trace out a circle, and you can also trace a circle with a pin, string and paper.  You can also make a lovely ellipse with two focal points with some string and some pins:

Why stop here, though? The definition of ellipse (let’s call it a 2-ellipse) can be expanded to include 3 focal points, 4 focal points, 176 focal points, really focal points, no matter what integer is.

n-ellipse: Take all the points whose distances to each of n focal points sum to a certain number. This is the n-ellipse.

James Clerk Maxwell was sixteen years old when he invented a contraption that generalized the pin and string method for constructing an ellipse. He figured out a clever way to wrap the string around so that you could use pins and string to produce a 3-ellipse.

## 3. Generalizing Equations

The equations generalize nicely too.

Circle (a.k.a. 1-Ellipse): $\sqrt{(x-a)^2+(y-b)^2} = D$

2-ellipse: $\sqrt{(x-a_0)^2+(y-b_0)^2} + \sqrt{(x-a_1)^2+(y-b_1)^2} = D$

3-ellipse: $\sqrt{(x-a_0)^2+(y-b_0)^2} + \sqrt{(x-a_1)^2+(y-b_1)^2} + \sqrt{(x-a_2)^2+(y-b_2)^2} = D$

n-ellipse: $\sum_{i=0}^{n}\sqrt{(x-a_i)^2+(y-b_i)^2} = D$

(Most students learn a different equation for an ellipse, defined in terms of minor and major axes. It’s not exactly a load of fun to analytically derive that from this equation for the 2-ellipse, I’ve found.)

## 4. Smallest Ellipses?

Each of the definitions above leave room for a “certain distance.” But are there any restrictions on what that distance can be?

For a circle, that “certain distance” is the radius, and the radius can be whatever magnitude that you like. Given a center, make your radius huge, make it tiny: there’s always a circle of those points just that distance away.

For a 2-ellipse, though, this isn’t so. Imagine one focal point that’s one million miles away from the other, and imagine that we’re interested in the points whose sum of distances from the focal lengths is one inch. That’s simply impossible to achieve. Being one inch away from one focal point would necessarily involve being nearly a million miles away from the other.

At some point, an ellipse might be too small for its focal points.

So: what is that least distance, given two focal points? What is the least amount of distance that you can sum to, given focal points? What are these critical points of the distance sum function?

For a 2-ellipse, the minimum distance should just be the distance between the two focal points.

But what about for a 3-ellipse? or a 4-ellipse? an n-ellipse?

I’m still having trouble with the analytic solution in that paper, but graphically you can just graph a bunch of different distances and try to see where things are headed. Here is a snapshot from this Desmos graph, which graphs a collection of distances for three foci (which you can move around the graph — try it, it’s fun!).

This is a contour plot of the distance-sum function. We could also visualize this as slices of a 3D graph, where the “z” axis is the varying distance-sum.

Here is a 3D graph of a 2-ellipse, showing how it bottoms out at a very narrow ellipse (practically a line segment) at a height of 2 — the distance between the two foci:

And here is a 3D graph of a 3-ellipse, which also bottoms out at a certain distance-sum above 0:

In contrast, here’s the 1-ellipse, a circle. It has no minimum radius:

## 5. A cool problem

Say that you have three cities that are special to you, for whatever reason. Maybe you grew up in one of them, went to college in another, and have some family in the third.

If your only priority was to be as close as possible to each city, where should you live?

This is really, really similar to the question of where the minimum distance-sum is for an n-ellipse.

(It makes sense that it starts getting circle-ish as we get farther away from the focal points. From a million miles up in the air, those focal points might as well be on top of each other.)

Go to this graph and play around with your cities. You can also add more cities.

## 7. Now what?

I’m not sure. I tried to write this a few different ways, but in the end all I could figure out was the info dump you see above. Is there a way to rewrite this so that there’s a stronger narrative? Should I come up with problems for each section to give readers a chance to stop and think about stuff?

I think this needs to go on the back-burner, and I just need to write more about math and hope that eventually I figure a few more things out.