Author: davidkeithbutler

Where the complex points are: on a parabola

This is the third in what has turned out to be a series on Where the Complex Points Are.

Where the complex points are: introduction to the iplane concept
Where the complex points are: on a line
Where the complex points are: on a parabola (YOU ARE HERE)
Where the complex points are: the graph of a function
Where the complex points are: where idea came from

In the introduction, I talked about how I’ve always wondered where the complex points are on a parabola, especially wondering where the x-axis intercepts are if the parabola doesn’t meet the parabola.

A set of coordinate axes with a parabola drawn in blue. The parabola has its lowest point on the y-axis at y=1. The equation y=x squared plus 1 is written next to the parabola. Handwritten in red writing at the bottom of the picture is the question "Where are the complex points?"

Then I described the place where I imagine the complex points are: at each point \((a,b)\) in the real plane, there is an iplane attached, which contains all the points \((a+ci,b+di)\). These points are arranged with the x-axis showing the imaginary part added to the x-coordinate and the y-axis showing the imaginary part added to the y-coordinate. I imagine them as being a transparent sheet attached to the point itself and being able to be flattened out to sit on top of the real plane.

A pair of coordinate axes with background in pink. The axes are labelled i sub x and i sub y. Various points are marked on the plane in blue. At the origin, the point is labelled (3,2). At the point marked 3 on the horizontal axis, the point is labelled (3+31,2). At the point marked 4 on the vertical axis, the point is labelled (3, 2+4i). Ohter points on neither axis are labelled similarly, all with real part 3 and 2.

Four pictures in progression. The first picture shows an x-y plane with a parabola. The second picture has a cellophane sheet all scrunched up at the origin. In the third picture, the cellophane sheet has been opened out partly. In the third picture, the sheet has been laid flat on top of the x-y-plane and on it there are the points (-i,0) and (i,0) marked.

In the second post, I talked about where the complex points are on a line. Now I’m finally ready to talk about where the complex points are on a parabola.

The points in each iplane

When I looked at lines with complex slope, I looked at each iplane individually and found out which points in that plane were part of the line. (There was one in each.) Let’s do the same for our parabola.

Consider a parabola with equation \(y = ax^2+ bx + c\) and look at the complex points in the iplane attached to the point \((p,q)\). That is, the points \((p + ri, q + si)\)…

\[\begin{aligned}
q+si &= a(p+ri)^2+b(p+ri)+c\\
&= a(p^2+2pri-r^2)+b(p+ri)+c\\
&= ap^2+2apri-ar^2+bp+bri+c\\
&= (ap^2+bp+c-ar^2)+(2ap+b)ri\end{aligned}\]

Equate real and imaginary parts:

\(q=ap^2+bp+c-ar^2\) and \(s=(2ap+b)r\).

From the first equation:

\[\begin{aligned}
ar^2 &= ap^2+bp+c-q\\
r^2 &= \frac{ap^2+bp+c-q}{a}\end{aligned}\]

So we have an expression for \(r^2\). Since \(r\) is a real number, we’ll get a different number of solutions depending on whether this value is negative, zero or positive. But let’s look closer, when is this number negative, zero or positive?

Well, \(ap^2 + bp + c\) is the y-coordinate of the point on the parabola with x-coordinate \(p\).
So \(ap^2 + bp + c – q\) is the vertical distance between the point \((p,q)\) and the parabola. If the point is below the parabola, this value is positive; if it’s above the parabola, this value is negative; and if it’s on the parabola, this value is zero. The value of the coefficient a being positive or negative will change whether the whole expression is positive or negative. But it will also change whether the parabola itself is upward or downward curving. So overall we get the following possibilities.

	Point \((p,q)\) below parabola	Point \((p,q)\) on parabola	Point \((p,q)\) above parabola
Parabola upward-curving	Two solutions for \(r\)	One solution for \(r\)	No solutions for \(r\)
Parabola downward-curving	No solutions for \(r\)	One solution for \(r\)	Two solutions for \(r\)

Since the second equation only has one \(r\), each solution for \(r\) will give one solution for \(s\). This means that we now know many points of the parabola there are in each iplane, depending on where the iplane is attached.

If the iplane is attached to a point inside the curve of the parabola, then there are no points of the parabola in the iplane.
If the iplane is attached to a point on the parabola, then this is the only point of the parabola in the iplane.
If the iplane is attached to a point outside the curve of the parabola, then there are two points of the parabola in the iplane.

This GeoGebra applet allows you to investigate the complex points on a parabola in the iplanes attached nearby.

A screenshot of a GeoGebra applet. There is a set of coordinate axes with a a parabola drawn in blue. In the top left corner is a place where you can enter the coefficients of the parabola's equation. There is an iplane represented by a pink square centred on a moveable gold dot. Currently the gold dot is below the parabola and the pink square contains two red points on opposite sides of the gold dot.

Collecting together iplane points

If you’ve played with that GeoGebra applet for a bit, you might have noticed that when you move the point where the iplane is attached, the complex points on the parabola move in interesting ways across the screen. It’s actually very interesting.

In a completely technical sense, the points we see in the iplanes in my GeoGebra applets aren’t really where they look like they are. They are up in iplane which only touches our real plane at the centre. It’s just that when we flatten a part of the iplane out and lay it on the real plane it’s easier to see the points. I’m going to make a new terminology to describe where the complex point looks like it is when you do this flattening out..,

Given a complex point, when the iplane it is in is flattened out to lay on top of the real plane, the real point where it appears to be is called the real image of the complex point. It’s quite straightforward to find the real image of a complex point directly from its coordinates: the complex point \((p+ri,q+si)\) has real image \((p+r,q+s)\).

Anyway, what happens when you take all iplanes attached along a vertical line and collect together the real images of the parabola’s complex points in those iplanes? (There’s got to be a way to define terminology to make it easier to say what I just said! But right now I can’t think of anything good.) You can investigate this question visually in this GeoGebra applet.

Look at that! The real images of the points in the iplanes attached to that vertical line form a parabola of their own! The line I’ve shown there is the axis of symmetry of my parabola, and it appears that this parabola-of-real-images is just the original parabola flipped upside-down.

Let’s do some calculations. Think about all the points \((x,y)\) that are the real images of the parabola’s complex points in the iplanes attached to the vertical real line \(x=p\). These complex points are \((p+ri,q+si)\) for some \(q\), \(r\), \(s\) and have real images \((x,y)=(p+r,q+s)\), where \(q=ap^2+bp+c-ar^2\) and \(s=(2ap+b)r\).

Now \((x,y)= (p+r,q+s)\),
so \(y=q+s = ap^2+bp+c-ar^2+(2ap+b)r\).

But \(x=p=r\), so \(r=x-p\),
which gives \(y=ap^2+bp+c-a(x-p)^2+(2ap+b)(x-p)\).

This is the equation of a parabola.

So given any vertical line, if you look in all the iplanes attached to the line and collect together the images of the complex points in those iplanes, you make a parabola! What if the line is the axis of symmetry of the parabola? That is, what if \(p = -b/(2a)\)? Then \(2ap+b=0\) and the equation becomes \(y=ap^2+bp+c-a(x-p)^2\).

This is the equation of a parabola with vertex \((p, ap^2 + bp + c)\) — the same vertex as our original parabola. Its leading coefficient is \(-a\) — as opposed to \(+a\) for the original parabola — so it’s the same shape as the original parabola, but upside-down.

What this means is, if you want to find the complex points where a parabola meets, say, the x-axis, just flip it upside-down at the vertex and see where this parabola meets the x-axis. These are the real images of the complex points of the parabola on the x-axis, and they are in the iplane attached to the point where the axis of symmetry meets the x-axis.

iplanes and tangents

Did you notice the second equation we got above? It said that when \((p+ri,q+si)\) is on the parabola then \(s = (2ap+b)r\). The number \(2ap+b\) is the slope of the tangent to the parabola at the point where \(x=p\). The equation \(s = (2ap+b)r\) is an equation of a line of this slope inside the iplane through the centre of the iplane. When the iplane is laid flat on the real plane, you get a line of slope \((2ap+b)\) through the point \((p,q)\). This means that the complex points of the parabola in the iplane at \((p,q)\) are both on a line parallel to the tangent to the parabola at the point where \(x=p\).

A screenshot of a GeoGebra graph. There is a set of coordinate axes with a a parabola drawn in blue. In the top left corner is a place where you can enter the coefficients of the parabola's equation. There is an iplane represented by a pink square centred on a gold dot. There is a vertical line drawn through the gold dot, with a black point where it meets the parabola. The tangent to the parabola at that point is drawn. A line parallel to that tangent is drawn through the gold dot. In the iplane there are two red points, which both lie on the line through the gold dot that is parallel to the tangent.

But wait, there’s more! We just went vertically to the parabola from our iplane centre point. What happens when we go vertically to the parabola from those two complex points in our iplane? (Well, from their real images anyway.) The real image of these complex points are \((p+r,q+s)\). The matching points on the parabola have x-coordinate \(x=p+r\). Let’s find the y-coordinate:

\[\begin{aligned}
y &= a(p+r)^2 + b(p+r)+c\\
&= a(p^2+2pr+r^2)+b(p+r)+c\\
&= ap^2 + 2apr+ar^2+bp+br+c\\
&= ap^2 + bp +c+2apr+br+ar^2\\
&= ap^2 + bp + c + (2ap+b)r + ar^2\\
&= ap^2 + bp+c – ar^2 + (2ap+b)r+2ar^2\\
&= q+s+2ar^2\end{aligned}\]

Now \(q+s\) is the y-coordinate of the real image of our complex point, and we already figured out above that \(ar^2\) is the vertical distance from the point \((p,q)\) to the parabola. So the y-coordinate is the y-coordinate of the real image of our complex point, plus twice the distance from the iplane centre to the parabola. That’s pretty cool. Even cooler is the fact that both complex points are the same vertical distance from their matching real parabola point, so the line joining these points is parallel to the line joining the complex points.

What this means is we now have a geometric way of finding the complex points on the parabola in the iplane at P=\((p,q)\):

Draw a vertical line from P to the parabola and mark the point on the parabola. Let’s call it A.
Draw the tangent to the parabola at the point A.
Copy the distance from A to P inside the parabola from A. Let’s call this new point B.
Draw a line through B parallel to the tangent at A and mark where it meets the parabola. Let’s call these points C and D.
Draw vertical lines through C and D.
Draw a line through P parallel to the tangent at A. Mark where it meets the vertical lines through C and D.

These final two points are the real images of the complex points on the parabola in the iplane at P. You can see this construction in this GeoGebra applet. Isn’t that cool?

It also means we can do a similar construction to find where a particular line meets the parabola. Obviously if it meets the parabola in real points then there they are in the picutre. If it has no real points on the parabola, then we know that all the complex points on that line have real images on the line too. We also know that the complex points of the parabola in any iplane are on a line parallel to a tangent of the parabola. This means we have this geometric way of finding the complex points where a line meets a parabola:

Find the tangent to the parabola parallel to your line. Mark the point where it meets the parabola A.
Draw a vertical line through A. Mark the point where it meets the original line P — this is the centre of the iplane we need.
Follow the construction above to find the complex points in the iplane at P.

If your line is a horizontal line, such as the x-axis, then this process is particularly simple because the only point with a horizontal tangent is the vertex.

Conclusion

So, in every iplane centred at a point outside the parabola there are two complex points on the parabola, and the real images of those points can be found via a geometric construction using tangents. Isn’t that totally awesome?

In the final post, I’ll talk about where this idea came from, because I think it’s totally worth mentioning.

8 August, 2016

Where the complex points are: on a line
This is the second in what has turned out to be a series on Where the Complex Points Are.
1. Where the complex points are: introduction to the iplane concept
2. Where the complex points are: on a line (YOU ARE HERE)
3. Where the complex points are: on a parabola
4. Where the complex points are: the graph of a function
5. Where the complex points are: where idea came from
In the introduction, I described the place where the complex points are: at each point \( (a,b)\) in the real plane, there is an iplane attached, which contains all the points \((a+ci,b+di)\). These points are arranged with the x-axis showing the imaginary part added to the x-coordinate and the y-axis showing the imaginary part added to the y-coordinate. I imagine them as being a transparent sheet attached to the point itself and being able to be flattened out to sit on top of the real plane.

In this post, I want to talk about where the complex points are on a line. This is the first reason why this representation of complex points is so very cool.

A real line

Every real line has an equation like \(ax + by = c\) for some real \(a\), \(b\), and \(c\). For example, \(2x + 9y = 7\), \(-3x + 5y=8\), \(x – 2y = 3\), \(x=4\) and \(y=5\). (I’ve used whole numbers here but there’s nothing stopping us using fractions or square roots or transcendental numbers like e.)

But we know that at every point there is an iplane attached which holds all the complex points. Which of these complex points are part of the line? First we need to remind ourselves that a point is on a line when it satisfies the equation of the line. That’s fundamentally what an equation is for: deciding if a point is part of a set or not.

Let’s consider the line \(x – 2y = 3\)…
- The point \((3,0)\) is on this line because \(3 – 2\times 0 = 3\).
- The point \((1,-1)\) is on this line because \(1 – 2\times (-1) = 1+2 = 3\).
- The point \((9,3)\) is on this line because \(9 – 2\times 3 = 9-6 = 3\).
- The point \((3+2i,i)\) is on this line because \(3+2i – 2\times i = 3+2i-2i = 3\).
- The point \((1+4i,-1+2i)\) is on this line because \(1+4i – 2\times(-1+2i) = 1 + 4i +2 – 4i = 3\).
- The point \((9-10i,3-5i)\) is on this line because \(9 -10i -2\times(3-5i) = 9 – 10i – 6 + 10i = 3\).
There is a pattern forming here: in order to get my complex point to be on the line, I need to make sure the imaginary parts are the right sizes so that they subtract away when I put them in the formula. The real parts have to create the number 3 I need as well, which means the real parts have to satisfy the original equation.

Let’s do this with algebra:

Suppose the point \((p+ri, q+si)\) is on the real line with equation \(ax+by=c\).

Then

\[\begin{aligned}
a(p+ri)+b(q+si) &= c\\
ap+ari+bq+bsi &= c\\
(ap+bq)+(ar+bs)i &= c+0i\end{aligned}\]

Equating real and imaginary parts gives:

\(ap + bq = c\) and \(ar+bs=0\).

So \((p,q)\) satisfies \(ax+by=c\)
and \((r,s)\) satisfies \(ax+by=0\).

What does this mean? If the point \((p+ri,q+si)\) is on the line with equation \(ax + by = c\), then the point \((p,q)\) has to be on the line. This means that all the complex points on the real line are in the iplanes at the points on the real line. That’s very pleasant — the complex points on this line are in some way attached to the line itself, because they’re in the iplanes attached to this line.

But it gets better! The complex point \((p+ri,q+si)\) is the point with coordinates \((r,s)\) in the iplane at \((p,q)\). And the point \((r,s)\) has to satisfy the equation \(ax + by = 0\). That means the point \((r,s)\) lies on a line parallel to our original line passing through the centre of the iplane. When you take the iplane at the point \((p,q)\) and lay it flat on the real plane, the complex points in that iplane that are part of the line will lie on top of the existing real line! This is because they are on a line parallel to this line passing through a point on this line.

So in our representation, the complex points on a real line aren’t just in the iplanes attached to the real line, they look like they are actually on the line! Isn’t that cool?!!

You can play around with the iplanes attached to real lines in the following GeoGebra applet .

A line with real slope

There are a whole lot of lines in the complex plane that aren’t real lines. The first kind I want to deal with is those with real slope. That is, their equations are of the form \(ax + by = c + fi\) for some non-real number \(c+fi\) and real numbers \(a\) and \(b\). This kind of line has no real points at all – it’s not possible to put real numbers into \(ax + by\) and get a non-real result. But where actually are all the non-real points on this line?

Suppose the point \((p+ri,q+si)\) is on the line with equation \(ax + by = c + fi\).

Then

\[\begin{aligned}
a(p+ri)+b(q+si) &= c+fi\\
ap+ari+bq+bsi &= c+fi\\
(ap+bq)+(ar+bs)i &= c+fi\end{aligned}\]

Equating real and imaginary parts gives:

\(ap + bq = c\) and \(ar+bs=f\).

So \((p,q)\) satisfies \(ax+by=c\)
and \((r,s)\) satisfies \(ax+by=f\).

So the points on the line with equation \(ax+by=c+fi\) are in the iplanes attached to the real line with equation \(ax+by=c\), and in each in each iplane they form a line parallel to this real line. In fact, if you look at where they are across all the iplanes, they all form the same line parallel to the real line.

You can investigate these kinds of lines using this GeoGebra applet .

Of particular interest for our purposes later on are the “vertical” complex lines with equations of the form x=fi. The points on this line are in the iplanes attached to the real line x=0 — that is, they’re in the iplanes attached to the y-axis. Inside each iplane they form a vertical line, which looks like it’s part of the real line x=f.

A line with unreal slope

The last kind of line is a line with unreal slope. It would have an equation like \((a+di)x + (b+ei)y = c+fi\). You’d need to make sure that \(a+di\) and \(b+ei\) weren’t related by a real multiple or you could divide by some complex number to produce real coefficients for the x and y — we’ve already covered lines like that. In fact, it must therefore be possible to arrange it so that the y-coefficient is 1. In that case, we might as well write it in “slope-intercept” form, which might be a bit easier to process. So let’s take a look at the line with equation \(y = (a+di)x + (c+fi)\).

Suppose the point \((p+ri,q+si)\) is on the line with equation \(y = (a+di)x + (c+fi)\) for \(d\neq 0\).

Then

\[\begin{aligned}
q+si &= (a+di)(p+ri)+ c+fi\\
&= ap + ari + dpi – dr +c+fi\\
&= (ap-dr+c)+(ar+dp+f)i \end{aligned}\]

Equating real and imaginary parts gives:

\(q=ap-dr+c\) and \(s=ar+dp+f\).

Hmm. That’s not particularly edifying. What does that mean? Why don’t we pick an iplane and see what the points on our line look like in this iplane:

Consider the points in the iplane at the specific point \((p,q)\):

\[\begin{aligned}
q &= ap -dr+c\\
dr &= ap – q + c\\
r &= \frac{ap-q+c}{d}\end{aligned}\]

This means that given \((p,q)\), we can find a unique value of \(r\). Substituting this into the other equation gives us a unique value of \(s\). That is, every iplane has exactly one point of this complex line! You can investigate what these points look like with this GeoGebra applet.

Note that there is in fact exactly one real point every line with complex slope. Using the notation above, it’s the point \(\left(-\frac{f}{d}, -\frac{af}{d}+c\right)\). It’s quite hard to find the point by trial and error in the GeoGebra applet!

Conclusion

So there you have it. Using iplanes, the lines with real slope actually look like lines, and the lines with unreal slope are a cloud of points with one in each iplane. In the next post, I’ll talk about parabolas…
5 August, 2016
Where the complex points are
When you first learn complex numbers, you find out that they give you ways to solve equations that were previously unsolvable. The classic example is the equation equation \(x^2 + 1 = 0\), which if you’re only using real numbers has no solutions, but with complex numbers has the solutions \(x=i\) and \(x=-i\).

As someone who likes to imagine the physical reality of everything, this has always caused me considerable difficulties. The equation \(x^2 + 1 = 0\) can be thought of as the equation that tells you where the parabola with equation \(y = x^2 + 1\) meets the x-axis.

Only the parabola with equation \(y = x^2 +1\)doesn’t meet the x-axis. If our complex number solutions are to be believed, then it meets the x-axis in the points \((i,0)\) and \((-i,0)\), but I certainly can’t see those points on my graph. Where are they?

Presumably there are a whole host of points with complex coordinates, which are points where various things meet that don’t look like they meet. These points must be somewhere, and they must be some place that is somehow related to the graphs I see in the real plane. But where is this place?

Well, about a week ago, I finally found the place where the complex points are!

(At this point I need to make a few things clear: I have made this up entirely on my own. I have not learned it from anyone else, or from reading anyone else’s work. I am not so vain as to believe that no-one has ever thought of this before, but as yet I can’t find search terms in Google that have shown me anything close. If you have heard of it before I’d love to know!)

Here’s the idea: at every point in the real plane, there are a real-plane’s-worth of complex points attached. The complex points attached to the real point \((a,b)\) are all of the form \((a+ci,b+di)\). That is, the real parts of the two coordinates are \((a,b)\). I imagine them as a plane, with x-axis showing what imaginary part we’ve added to the x-coordinate, and y-axis showing what imaginary part we’ve added to the y-coordinate.The real point itself is in the centre of this plane.

I’m going to call this plane attached to the point \((a,b)\) the “iplane at \((a,b)\)”. In the picture below, I’ve shown a few points in the iplane at \((3,2)\). I’ve coloured it pink so you can tell it’s not the ordinary real plane, which as you saw above, I usually colour in white.

When I say “attached” I mean it literally — I actually do imagine the iplane being attached to the point in a physical sense, like in the photos here, which are at different levels of spread-out-ness:

Of course, every point in the real plane has its own iplane, so the complex plane as a whole looks more like this:

And that’s where the complex points are. The complex point \((3+i,5-2i)\) is in the iplane attached to the real point \((3,5)\); the complex point \((2i, -6+7i)\) is in the iplane attached to the real point \((0,-6)\); and the complex points \((i,0)\) and \((-i,0)\) are both in the iplane attached to the real point \((0,0)\)

It’s easier to see them if you spread the appropriate iplane out and flatten it so it sits on top of the real plane. You don’t want to do the whole plane of course; you just want a big enough slice so that you can see where the point is that you want. Also, I imagine the iplane being transparent so you can see the real plane through it — hence the cellophane in all the photos of the physical models.

In the below picture, a slice of the iplane at \((0,0)\) is being flattened out to show the points \((i,0)\) and \((-i,0)\).

This GeoGebra applet allows you to enter the coordinates of a complex point and it will show you the iplane it’s in and the real point where the iplane is attached. Give it a go to get a feel for what’s happening here.

So why am I so very very excited by this idea? Well, it all has to do with what happens when you figure out the complex points that are part of some familiar graphs. So far, I’ve thought a lot about lines and a bit about parabolas. It will take me a while to write about it all, so I’ll save those topics for the subsequent blog posts.

The titles of the five posts in the series are:
1. Where the complex points are (5 August 2016)
2. Where the complex points are: on a line (5 August 2016)
3. Where the complex points are: on a parabola (8 August 2016)
4. Where the complex points are: the graph of a function (10 August 2016)
5. Where the complex points are: where the idea came from (23 August 2016)
And then two more posts were done several years later:
1. Where the complex points are: on a complex line (again) (3 May 2020)
2. Where the complex points are: on a real circle (31 August 2020)
UPDATE: There was a later series of blog posts in 2022 where I modified the idea from iplanes to i-arrows, and an even later blog post in 2024 further investigating the line joining two complex points using i-arrows.

These comments were left on the original blog post:

Mark Pettyjohn 25 January 2017

When I saw “the biggest maths idea I’ve ever had” I didn’t think I had a prayer of understanding it. That I do is a testament to your understanding of the problem and solution and your ability to articulate it so clearly.

This post inspired a lot of thoughts that I tried to condense into something coherent here. I look at your “biggest idea ever” through the lens of the Standards for Mathematical Practice we use in the states k-12. The broader ideas are looking at and contrasting what mathematicians (like you!) do and what school mathematics is often like.

Thanks for the inspiration!

My blog post

Robert Low 16 April 2017

You’ve invented a way of visualizing R^4 by attaching an R^2 to each point of R^2, so in some sense the tangent bundle of R^2. It’s pretty neat, though, how you’ve made it concrete!

David Butler 18 April 2017

I’ve never known what a tangent bundle is before now, so I’m glad if I ever look into it more closely I will have something to work from!

Martin Flashman 20 July 2018

An alternative to understanding functions (of either real or complex variable) is to use mapping diagrams.
For quadratic functions (with either real or complex coefficients) the roots of the quadratic equations are visualized by arrow(s) from point(s) in the source complex plane to the point 0+0i in the target plane.
See for example my work with GeoGebra https://www.geogebra.org/m/JZYgq4Mx#material/aYeGJpS4 and MD to visualize solving linear and quadratic equations in https://www.geogebra.org/m/CX32kuXG#chapter/274869 .

David Butler 24 July 2018

Yes I am aware of this sort of thing and it’s an extremely useful way to think about things. I find it especially useful for derivatives. Thanks for sharing!
5 August, 2016
Brackets

I had a meeting with an international student in the MLC on Friday who has having a whole lot of language issues in her maths class.

She was from the USA.

Yes, the USA. Her problem wasn’t the everyday English; it was with the different terminologies for mathematical things here compared to her experience where she comes from. Her only experience with vectors was in physics where a vector is a quantity with magnitude and direction, whereas in maths class here, a vector is usually just a list of coordinates. She knew how to find a derivative, but had never heard the word “differentiate” used for that action. She had only ever used the word “anti-derivative” for what we call an “integral” most of the time.

I was talking through the transpose of a matrix (something else new to her) and how it interacts with other operations on matrices, and how since it’s normally written as a power it takes precedence over nearby multiplications like powers do. She asked me if, here in Australia, we still use the same order for what is supposed to be done before other things. I said, yes, we do, and told her that most local high-school students use the acronym BEDMAS to describe that order.

She wrote underneath the acronym familiar to her: PEMDAS. First she focussed on the fact that one had DM and the other MD, but reconciled that quickly saying, “Well I suppose they go in the order they come and so it doesn’t matter which way around they are.” But she had no idea what to do with the B.

I told her the B stood for brackets, and I drew what brackets look like: ( ), [ ], { }. And then she freaked out. To her, those things are called parentheses ( ), brackets [ ] and braces { }. I said, yes, those are the official titles of those things, but here in South Australia they’re thought of as different kinds of brackets. If we want to distinguish between them we’ll call them round brackets ( ), square brackets [ ] and curly brackets { }.

And suddenly a light came on for her and a whole lot of stuff people had said this week made sense. She also suddenly understood the very odd look her class tutor gave her when she mentioned the word parentheses. “Yes,” I responded. “Most maths students and tutors here would never have heard the word parentheses.”

And what happened next? Well we’ll have to wait and see. I think we made some excellent progress, and we agreed to keep meeting across the semester to help deal with anything more that might come up.

For me, I’m so glad I knew a little about the differences between Australian and American mathematical-English. (Thanks MTBoS!) And perhaps if anyone is reading this, then you will know too.

Australia USA
( ) round brackets parentheses
[ ] square brackets brackets
{ } curly brackets braces

PS: I find it interesting how the Australian acronym BEDMAS references a general term “brackets” which covers all shapes of bracket, whereas the American acronym PEMDAS references a specific term “parentheses” which only covers one shape of bracket.

These comments were left on the original blog post:

Claire 31 July 2016

This is very interesting. I teach high school in Southern California. When I teach the Order of Operations, my students have seen it before from middle school. They were either taught PEMDAS or Please Excuse My Dear Aunt Sally to learn the order. I always use these acronyms as a way to highlight the limitations of some of the “tricks” they use to memorize math concepts.

It’s interesting that the parenthesis, brackets, and braces are all called “brackets” where you are. Having 1 word for that is super helpful. I tend to say that they are all “grouping symbols” and if it were up to me, the acronym would be GEMDAS. Because then, for example, absolute value symbols are grouping symbols.

We also discuss why MD doesn’t matter the order… That they are inverse operations and part of a mathematical family. Same with AS.

This family idea is helpful to explain where roots and logs might go in the order of operations.

This post gave me a lot to think about.

Geoff Coates 1 August 2016

I helped an African student once who learned his high school maths in French. They used commas where we use decimal points and full stops where we use commas in large numbers. He was very confused for a while …

David Butler 2 August 2016

This came up in the Chemistry labs last semester, when all the pipettes had their volume listed as “0,25mL”.

31 July, 2016
TMC16 reflections from someone who wasn’t there

This post is about my response to TMC16.

For the uninitiated, TMC is short for Twitter Math Camp. This is a conference designed by teachers for teachers with teacher speakers, organised through the collective efforts of the Math Twitter Blog-o-Sphere (MTBoS) – a group of people who blog and tweet about their experiences teaching math(s). That description is not the best description of the MTBoS, but I’ll get to that later.

For those who were actually at TMC16, you may be thinking, “But David, you weren’t at TMC16!” And I wasn’t, not physically at least. But then Tracy Zager mentioned me in her keynote, and Megan Schmidt told me that she did, and suddenly I was drawn into a most interesting world I had never observed before.

Over the next week, I followed the #TMC16 hashtag on Twitter as people live-tweeted sessions, live-recorded them and posted them on YouTube. I participated via Twitter in the activities they did during sessions, and in discussion in the days afterwards. I read any number of blogs about people’s experiences attending.

Now I have so many thoughts spiralling through my head, and I need to write about them to process them. I also need to put TMC16 to bed because semester starts again today and I have so many jobs to do, some of which I didn’t do during the mid-year break because I was virtually attending TMC. So here goes: my TMC16 reflections.

The conference structure you wish you had

The conference itself sounds like the best setup for a conference I have ever known: The morning sessions were all connected across the three days to allow for focussed development of ideas. There was one keynote speaker a day. There are afternoon sessions based on accepted proposals, with one more timeslot on the last day for people who weren’t accepted to do sessions of their own (called “Flex” sessions) [Note: see Lisa’s comment below for what Flex sessions are actually designed for.]. Anyone at the conference could sign up to do “Favourites” talks of up to 15 mins which were done at whole-conference sessions at various points during each day — literally anyone from the newcomer to the seasoned veteran. Plus there was a full 90 minutes for lunch!

I have never experienced a conference with such a diverse set of ways to learn and interact: three-day workshops, short sessions, long talks, short talks, long lunch. I have also never experienced a conference with such an explicit focus on encouraging community interaction. At what other conference has a newcomer ever been encouraged to actually do a 10-minute talk to the whole conference, and cheered on when they do so?

I have been to several conferences in the past and felt like an outsider. I’ve felt like others there were concerned with meeting up with their once-a-year friends more than welcoming newcomers. I’ve felt like somehow you have to be better than you are to be allowed to participate or have a voice. Watching the Favourites talks of people, many of whom were newcomers, some of whom revealed quite personal stories, made me wish for something better. Reading people’s thoughts about how welcoming everyone was to their conversations made me wish for something better.

Keynote speakers who are honest, encourage you to think and then do something about it

There were three keynote speakers across the three days, chosen from among those in the MTBoS: Dylan Kane, Jose Vilson, and Tracy Zager. Each of them talked with humour, passion, compassion and honesty. They brought their perspectives on what teaching can be about, what the MTBoS can be about, and then had calls to action to encourage the listeners to do something about what they had heard. This is a brilliant concept and really brings home the idea that the conference wasn’t just a love-in and a place to passively receive information, but is supposed to empower others to change.

What I particularly appreciated was the three speakers’ honesty. Each of them told their own journeys and struggles with teaching and participating in this community. It’s a different type of thing from keynote speakers I’ve seen before, who are called in as experts and always seem to have all the answers. These three on the other hand clearly admitted their own shortcomings and how they still have a lot to learn. It was so refreshing.

A paradox: more than just resources, but was it ever just resources?

Dylan in his keynote mentioned how the MTBoS needs to become more than resources. He noted how when he was a new teacher, he was a magpie bringing in shiny new resources and activities into his classroom from the MTBoS and “throwing them at the wall to see what would stick”. It was a long time before he realised that what he needed was ideas that underpin these to make them work in his own context. He encouraged the MTBoS to be more than just a collection of resources people have made.

Tracy in her keynote mentioned how she as an early school teacher was intimidated by the number of high-school teachers in the MTBoS discussing on Twitter, and how she overcame this to become part of these conversations. The point is that she seemed to see the MTBoS as a conversation to be joined.

I find this contrast most interesting. I’ve been watching the MTBoS for about a year, and I never had the impression of it as a list of resources at all! Only now am I starting to find these resources that Dylan mentioned. Perhaps this is because it was introduced to me as a Twitter discussion hashtag first. I saw it as a place where people asked questions and in response got various answers and mostly thoughtful discussion. That’s what I was seeing on Twitter. When I followed links to people’s blogs, I mostly saw discussion of how they had tried to implement things, or thoughtful pieces about what teaching is and could be. Very rarely did I see a blog post simply listing resources people had made, and if I did I never found them that inspiring, so I didn’t go back there.

In light of Dylan’s talk, I realised I had somehow selected my experience of the MTBoS to be about the discussion and not the resources. Another reason for this self-selection may be because I am not a classroom teacher any more and so it’s not the resources that I think I need. I can’t use resources in my teaching most of the time because of the teaching situation I am in, so I gravitated to the discussion. That discussion fed my thinking about teaching, rather than some other more physical need.

I have to say that I agree with Dylan that there needs to be more about the thinking when people do post resources. As a new teacher in a school where I was it for maths and science for the whole school, I really needed that thought process to make sure I actually went through a thought process, rather than “throwing to see what sticks”. As a teacher in university looking at resources that are at the school level, I need the thinking to help me sort out if and how I could apply this stuff to my own teaching situation. I suspect Tracy would agree that the thinking would allow an elementary teacher to relate things talked about by a high-school teacher to their situation, too.

The really interesting thing is that TMC itself shifts the focus from resources to discussion. I can’t count the number of TMC recaps that have mentioned the fact that going to TMC and interacting with people in person made them feel more part of the community and therefore more able to engage with discussion. I can’t count the number that said that it was not the resources and activity ideas they appreciated but the chance to thrash out their thinking about teaching. I think there’s a lesson there about what the MTBoS is really about.

Post-TMC malaise

One problem with watching but not attending TMC (and possibly the same for attendees too) is that it only heightens my sense of disconnectedness at home. My teaching situation is quite rare, and it can be hard to feel connected. Also, at my university at least, not everyone is willing to have discussions about teaching at all, and sometimes only willing to do so if there is a rigorous research backing first, rather than the refreshing can-do attitude I see in the members of the MTBoS. Even when I do have them, it’s hard to feel like it’s a valid use of my time when there’s the huge pile of mostly administrative jobs that have to be done. I don’t know if there’s anything I can do about this.

Conclusion

Thanks to Megan and Tracy for inviting me in to the TMC experience. Despite the malaise, I count it as a most rewarding experience. Oh, and I gather I am supposed to choose one TMC thing that I plan to work on for the next year. Well, I choose to be more active in welcoming people to the teaching discussion. I want the teachers around me to know that I really care about their thoughts and want to learn from them. I choose to ask others for their thoughts on teaching and to really listen.

So there’s my TMC16 reflections, from someone who didn’t actually go.

These comments were left on the original blog post:

Lisa Henry 25 July 2016

Hi David!

Thank you for taking the time to blog about your experience with #TMC16 from the perspective of someone who attended virtually. It was really interesting to read what you gleaned from all of the tweets.

For those of David’s readers who don’t know me, I am the lead organizer of TMC16 and have organized it from the beginning.

I did want to offer a couple of clarifications to the TMC experience. I have to say, I am incredibly impressed how much of the TMC experience you have correct for never having attended one and just learning of it from being mentioned in Tracy Zager’s keynote! “Flex” Sessions (on the last full day) actually serve two purposes: one, if someone has something they would like to present that came up after the original speaker proposal deadline in January, or two (and this was the original purpose), as conversations happen over the course of a couple of days, people wish to spend time discussing it further. Sometimes what happens is people ask for a session to be repeated or for a different focus. For example, last year, Chris Harris gave a 30 minute session on Number Talks at the Elementary level and so many people asked her about how to do Number Talks at the Secondary level that she repeated it as a Flex Session with that focus.

You said: “I have also never experienced a conference with such an explicit focus on encouraging community interaction.” That is very much what the focus is for TMC. As much as I’d like to say it’s about the learning, it is really (for me) about the interaction with everyone. We started TMC in 2012 as an extension of our interactions on Twitter. We needed to meet in person. The face-to-face interaction was, and still is, what drives me to continue to organize this conference each year.

I am glad that you shared your perspective. It helps me to see other’s perceptions of what happens at TMC. Maybe at some point, you’ll be able to attend one in person. Next year, we’ll be outside of Atlanta, Georgia from July 27-30, 2017. Maybe then?

All the best,
Lisa Henry
Chief Organizing Officer, TMathC
Lead Organizer, TMC16

David Butler 25 July 2016

Thank you for your comprehensive response Lisa!

I’m glad I got the overall impression right from my first interaction with TMC, especially only having heard of it this year. I have to say it’s the live tweeting and the people’s blogs that make it clearer what it’s about, more than the website itself.

I may never come to a TMC because travelling outside Aus is a big deal for me, both for work and family reasons. But I will certainly follow along online each year in July!

UPDATE: I actually went to TMC the following year and you can read all about it in my TMC17 diary post.

25 July, 2016
David Butler and the Prisoner of Alhazen

Once upon a time, I did a PhD in projective geometry. It was all about objects called quadrals (a word I made up) – ovals, ovoids, conics, quadrics and their cones – and the lines associated with them – tangents, secants, external lines, generator lines. During the first two years, I did talks about my PhD research, which I could not resist calling “David Butler and the Philosopher’s Cone” and “David Butler and the Chamber of Secants”.

At that time, my use of JK Rowling’s titles had to stop because there was no suitable mathematical thing to insert into the third title. It’s been ten long years since “David Butler and the Chamber of Secants”, and finally I have found something to use. Hence, welcome to…

David Butler and the Prisoner of Alhazen

Chapter 1: In which David finds out about lunes for the first time

As you may have noticed, I have quite a love affair with the Quarter the Cross problem, having blogged about it here and here. This blog post is not about Quarter the Cross, but the story did start there.

At the December holiday One Hundred Factorial session, we did Quarter the Cross together, and these were some of the solutions we came up with:

You may notice some rather interesting solutions involving curved shapes. Well, one of the students (Adam) created the first of these, and of course we all asked him how he was doing it. He told us the shapes were called “lunes” and showed us how to draw them: you draw semicircles outside the short sides of the triangle, and then one big semicircle inside the longest side, which has to go through the point because semicircles on hypotenuses always pass through right angles. Then he declared that the area of the two lunes was the same as the area of the triangle. (That is, in the following triangle, the striped area is the same as the pink area.)

Of course we needed a proof of this, and we duly worked together to construct one. Here’s a pretty version of the proof we came up with:

The calculations in the above proof are as follows:

\[\begin{aligned}
\text{Area of big semicircle } &= \frac12 \pi \left(\frac{c}{2}\right)^2 = \frac{\pi c^2}{8}\\
\text{Area of middle-sized semicircle } &=\frac12 \pi \left(\frac{b}{2}\right)^2 = \frac{\pi c^2}{8}\\
\text{Area of small semicircle } &=\frac12 \pi \left(\frac{a}{2}\right)^2 = \frac{\pi c^2}{8}\\
\text{Area of entire diagram } &= \frac{\pi a^2}{8}+\frac{\pi b^2}{8}+\text{ Area of triangle}\\
\text{Area of two lunes } &= \left(\frac{\pi a^2}{8}+\frac{\pi b^2}{8}+\text{ Area of triangle}\right)-\frac{\pi c^2}{8}\\
&= \frac{\pi a^2}{8}+\frac{\pi b^2}{8}-\frac{\pi c^2}{8}+\text{ Area of triangle}\\
&= \frac{\pi}{8}(a^2+b^2-c^2)+\text{ Area of triangle}\\
&= \frac{\pi}{8}\times 0+\text{ Area of triangle}\\
&= \text{ Area of triangle}\end{aligned}\]

Later that day, I tweeted about this, and someone asked what this theorem was called. I didn’t know, so I went looking online for more about these things. My first problem was that I was searching for “loons” instead of “lunes”. Afterwards, it made perfect sense that they were called “lunes”, since “lune” is the French word for “moon”. But when I was searching, I had only heard the word spoken aloud and not seen it written down, so I couldn’t know that it was actually a French word could I? After what seemed like hours, I finally did find it. I was not disappointed.

These things have the coolest name of any maths object I know, worthy of any Harry Potter title: the Lunes of Alhazen! They are named after an Arab mathematician/scientist/philosopher who lived about a thousand years ago, and was once so famous that if you simply said “The Physicist” then people knew you were talking about Alhazen. (Funny how most of us educated in Western countries have never heard of him.)

Chapter 2: In which David is imprisoned by Alhazen’s lunes in two different ways

But the story didn’t stop there. As with most very cool maths things, it sort of takes over a corner of your mind, holding you prisoner, as it were. This Alhazen and his lunes held me and others prisoner in various different ways.

One way I was held prisoner was that I found myself trying to see lunes everywhere. Every time I saw a right-angled triangle in a design, I’d wonder about the two lunes you could make from it. MLC staffer Damien confessed this same obsession to me not long after the One Hundred Factorial session, except that he was actually carrying a pair of compasses around to do the actual construction. In Quarter the Cross itself, I had quite an ambition to find a single lune that had an area of a quarter of the cross. I did find it, and here it is:

(Technically, this lune is the Lune of Hippocrates. You can make two of them as lunes of Alhazen on the short sides of an isosceles right-angled triangle, but one big one can be drawn on the hypotenuse too, to give the same area.)

The other way the Lunes of Alhazen held me prisoner began when I realised that our earlier proof didn’t rely on the fact that the shapes I had drawn were semicircles. I could use any shape as long as they were all similar to each other, because the areas of similar shapes are always proportional to the squares of matching lengths. So I started thinking about the concept of the “Things of Alhazen” – objects constructed in this way from a right-angled triangle. Here are a few:

The concave hexagons of Alhazen

The connected triangles of Alhazen

The strange shapes of Alhazen

And a few more, after I realised they could go inside the triangle too.

The dolphins of Alhazen

The loons of Alhazen

And a few more, after I realised the big shape on the hypotenuse didn’t technically have to go through the right-angle.

The big concave hexagon of Alhazen

The almost lune of Alhazen

(Note they’re all polygons because I was doing them in Geogebra and it was the easiest way to colour things in the way I wanted. I suppose I could have done it in Geogebra and copied the picture and bucket-filled, or drawn them in some other program. But this is what I did.)

Chapter 3: In which Tracy is also a prisoner of Alhazen

But I wasn’t the only one who was imprisoned by these lunes of Alhazen. This week it was “Twitter Math Camp”: a conference where those in the MTBoS come together to share their reflections and learning, organised by teachers for teachers. At TMC, Tracy Zager did a keynote talk where she talked about the benefits that happen when people at all different levels of education come together to learn about teaching maths from each other. (You can see the part of the video directly related to this story, and see the slides or give comments to Tracy at her blog.)

Tracy told the story of Quarter the Cross, and how it leaped up and down the levels of education and back and forth around the world. When she originally saw the work me and my students did, she was instantly caught by the curved shapes, and wanted to know more. Upon hearing the name of them, this was her response:

She was instantly captured by the wonder and the history and the awe, just as I was. But the imprisonment is deeper than this.

That tweet was seven months ago, and yet here is Tracy in a keynote talk at a conference, remembering this as a key moment in her journey as a maths educator. In a way, Alhazen has captured a part of her memory. She will forever think back to this time and have it all wrapped up in these lunes. And so will I.

We are prisoners of Alhazen in the best possible sense.

These comments were left on the original blog post:

Nic Petty 20 July 2016

Thanks for this. These are so cool they are almost magical.

Mardalee Burwitz 10 August 2016

Earlier today I was catching up with the #TMC16 and viewed Tracy Zager’s Keynote Video. The Lunes of Alhazen really jumped out at me. I am excited to bring this to my students this year, along with the Quarter Cross. I believe that after this past summer’s negative incidents, and our coming together in Rio this August, I feel as an educator to stretch and find connections btwn mathematics derived within as all cultures. The Lunes will work in great! Thank you for sharing.

David Butler 10 August 2016

Glad you can use them! I’d love to hear how it goes when the time comes.

19 July, 2016
The TMC Attitude

I think this will be my last post about Twitter Math Camp (TMC), getting in just before the TMC18 officially starts (though a lot of people are already there tweeting their TMC-eve adventures even as I write).

TMC is a truly remarkable conference, as I have described before, both in 2016 when I wasn’t there, and in 2017, when I was there. You can click on those links to read what I thought about it back then. The short version is I have never seen or experienced anything quite like it, with such a dense concentration of professional learning and networking, and also making deep connections with other passionate people.

In this last TMC17 post, I want to talk about the thing that I think makes TMC so special: the attitude. You see, other conferences might be able to have multiple different ways to share and learn from each other, and they might have opportunities for making real connections with others, but I think they might still fall short without this one special sauce that is the TMC attitude. One way to describe it would be to say the people at TMC come open and eager to be part of the thinking and the community.

The first aspect of this TMC attitude is that everyone in every face-to-face session is there ready to learn something. Not just expecting it but looking out for it — they are actively seeking for something to learn. More than this, not one person I saw at TMC last year was expecting to have something handed to them by a presenter, as if the presenter was some ordained expert, but everyone was fully expecting to process what they had learned through ongoing discussion with their colleagues, which would happen at breaks and late into the night. Even more remarkable than this, the presenters themselves all seemed to expect to learn something in the sessions that they themselves were presenting! (I know I certainly was expecting this.)

You may be thinking that at some other conference of course you are there looking for something to learn, and sure you are. But if I’m honest, sometimes I don’t know if I can go to yet another session that seems to be selling me a product, and sometimes I do go into a PD (Professional Development) expecting someone to just hand me the content without me needing to process it. Some PD experiences are almost designed to be about an expert handing down their product or policy and you don’t hope or expect to think. At TMC you almost always go in with an active attitude.

The second aspect of the TMC attitude is that everyone is there ready to be part of the community. As a presenter, everyone who comes to your session is a colleague you can discuss ideas with later outside the session. As a participant, you know that the presenter is another colleague who has something they are excited about to share and you want to be a support to them. You also feel that everyone around you is on the same journey, hoping to learn something from each other. Outside of sessions there is always someone to talk to and listen to and share both work things and life things with, and you know you can keep up a connection with these people through the medium of Twitter beyond the conference. Even if you don’t feel like engaging with people, you can let it all wash over you and feel part of something bigger.

Other conferences I have been to don’t have the same community feel. Some have been close, but not nearly at the same level. Certainly it’s rare for the presenter of a keynote to be around at the rest of the conference to be part of the community and discuss other people’s ideas, and rare for the audience to act like they are there to support them! It’s certainly true that no compulsory PD I’ve ever been to has had this attitude of community-building! At TMC community-building is always in the background.

I think that the attitude of the people at TMC is what makes TMC so remarkable.

But it doesn’t just have to exist at TMC — I think I can take this attitude to other PDs I go to in order to make them better. I too can go to sessions expecting to be actively learning something. and I too can go to sessions actively seeking to make real connections with people, especially in any sessions I present myself. If all of us do this more, then all PDs might be just that bit more like the awesomeness that is TMC.

I wish the TMC attitude for all of you.

18 July, 2016
Mansplaining

A few months ago, I learned a new word: “mansplaining”. You may have heard it before, but I never had until this year.

The general idea is that very often, a man will explain something to a woman in a way that seems to be based on the assumption that the woman is incapable of understanding the concept themselves and requires the man to rescue them from their misunderstanding. Often it is very explicitly patronising or condescending. This is a mansplanation.

In recent weeks, I have seen people I greatly respect being treated this way in the online space, and they have called out the man in question by telling him that he was mansplaining. Quite often, he has responded with quite a bit of vitriol, claiming that the word “mansplaining” is in itself sexist and they were just “trying to help”. This very vitriol is of course really not supporting the man’s case, and tends to show that his assumptions actually are that the woman did need to be rescued from her ignorant state. You can see some classic examples of this sort of assumption in Fawn Nguyen’s excellent blog post “Baklava and Euler “.

I had formed the idea that mansplaining was really just assholesmanplaining, and it didn’t have all that much to do with your general everyday respectful man.

But then something happened that hit me in the guts. Megan Schmidt started a conversation on Twitter about notation, and it had a flurry of responses, all from men, one of whom was me. She tweeted separately that “the mansplaining game is strong right now”. I was not consciously responding from an assumption that Megan needed to be rescued from confusion, and yet the conversation was called mansplaining. Clearly Megan’s use of the word didn’t fit with my understanding that only assholes mansplain.

It was time to get to the bottom of this, so I asked Megan to help me understand what she meant and how she felt about it. I have to thank her a hundred times for the thoughtful and gracious responses that she gave. I hope I will do justice to what you taught me, Megan!

I learned that there are times when offering an explanation at all is actually mansplaining. Not because the explainer is an asshole, or because they meant to be condescending or sexist, but because the explainer is unwittingly playing to a wider cultural assumption that the woman needs an explanation at all.

When a woman expresses frustration or anger or worry at something, a man’s common response is to offer an explanation to clear up confusion. Do you see the disconnect there? The man is rescuing the woman from confusion, but the woman wasn’t expressing confusion. She didn’t need an explanation – she didn’t need to be “rescued”. It’s most likely that she actually does understand the nuances of the concepts involved. Indeed, she would usually have to understand in order to have the emotional response she is having.

An unfortunate part of it is that the majority of men in this situation, especially in a professional setting, actually do realise that the woman does have the same or greater experience and training. It’s just that they are culturally conditioned to offer explanations in response to frustration. Indeed, it seems to be that men in professional settings are expected to engage in more “academic” conversations than “emotional” ones. Yet by doing so, we are still mansplaining.

The problem is that it opens the door for assholemansplanations, which are sure to follow. Even worse, it is adding to the hundreds of tiny sexist events that occur for a woman every day. And it reinforces the very cultural norm that produces those daily tiny sexist events. It’s important to give the experience a loaded name like mansplaining to make sure that those of us who do care have our attention drawn to these problems.

But how, as a man, can I fight back? Well, I can certainly call out others when they are mansplaining. Assholemen need to hear it from other men to have a chance of hearing the message – they’ll never listen to a woman. Ordinary men need to know about the damage they do unintentionally.

And what about my own daily actions? All I can think of is to be more aware. I can listen to the actual words people are saying and notice the emotional part of what they say. I can choose to respond by asking for more information first, rather than launching into an unwanted and unnecessary explanation. It takes a lot of energy to watch your own words and actions, and sometimes I will slip (sorry in advance) but with practice I’ll get better at it. And then one day maybe I’ll find I never offer a mansplanation again.

15 July, 2016
TMC crochet coral quietness

Here is another blog post in my series of only-a-year-late posts about Twitter Math Camp 2017 (TMC17). In this one I want to talk about the Crochet Coral workshops Megan and I did, but I don’t want to actually talk about the crochet coral. Instead I want to talk about the quietness.

TMC was a wild wild time. At the sessions there were hundreds of amazing maths and teaching ideas to process. At break times during the conference there were things to play with and discuss, with overheard conversations going on all around me about interesting stuff. At the social time outside the conference there were things to play with and discuss, with overheard conversations (and singing) going all around me about interesting stuff. It was exciting and wonderful, and mentally noisy.

Our three morning sessions were an island of calm and quiet in this tumultuous sea.

At our first session, Megan and I got everyone started on how the iterative crochet works, and from that point forward we explored where these simple ideas could take us. Somehow it produced six hours of wonderful calm.

We sat and focused on our own fluffy explorations, or quietly played with the existing corals, or clamly discussed the events of the conference so far or other thoughts that were flitting through our heads. Most of us had the conference Twitter feed on to look at, but it was almost as if it was outside us and we were observing from afar. In that room we were somehow created a protected space where we could recharge our energy to engage in the wild ride that was the rest of TMC.

I think there were two features of the activity we were doing that helped to create this magical space. The first is the slowness of crochet exploration. In other mathematical domains, exploration can be a chaotic whirl of sound and colour, with frenzied scribbling and conversation. But the exploration in crochet is slow and can take hours to get to the stage where the structure is visible in the thing you are making. The second is the smallness of the actions you perform to do the exploration. The mechanism you use to produce these corals uses small stitches that hold your attention in a tiny bubble between the tips of your fingers, almost causing the rest of the world to fade into the background. Joey described the strange feeling of looking up and finding how big the world is, after you’ve spent the last half an hour looking at the tip of a crochet hook.

I didn’t mean for our workshop to have this haven-like property but I am so glad it did. It really made an amazing positive difference my experience of TMC, and I don’t think I would have had nearly as wonderful a time without it. By the third day I was aching to get there to be part of our little crochet cuddle. If I ever go to TMC again I would want to find or make a similar space to have those quiet times. In the year since, I think I have forgotten how important that quiet time of focus is, and I think in the next several months I will try to recapture it again.

Thank you to the wonderful crochet crew for the quietness.

15 July, 2016
The line at infinity: conics

The story so far

I promised Tina on Twitter that I would write about how the line at infinity relates to conics, and I’ve been doing it in the last two blog posts.

First, I talked about what the line at infinity is. We noticed that a set of parallel lines all share a slope, so we elevated all the slopes to the status of points and gave them a more physical reality as “points at infinity”. Now lines we thought were “parallel” aren’t anymore because they actually meet. Any two lines meet, though possibly at a point at infinity.

Second, I talked about a system of coordinates that includes the line at infinity. It turned out we could do this by giving both points and lines three coordinates, with the understanding that any nonzero multiple of a set of coordinates represents the same point/line.

These coordinates are called homogeneous coordinates and they go like this: Points have coordinates of the form (x,y,w). Ordinary points have w not equal to zero, and their original coordinates can be calculated as (x/w, y/w). Points at infinity have w equal to zero, and a point of the form (a,b,0) is the point at infinity belonging to all the lines parallel to bx – ay = 0. Line equations go from ax + by + c = 0 to ax + by + cw = 0, which can be shortened to coordinates [a,b,c]. The line at infinity has equation w=0.

New perspectives from homogeneous coordinates

Last time I noted how awesome it is that with these new coordinates, lines and points are not really distinguishable from each other, if we just look at their coordinates. There is another thing I think is very cool too: with these new coordinates, the points at infinity don’t look fundamentally different from ordinary points. So what if one of their coordinates is zero? Other points have a zero coordinate, such as all the points on the x-axis — they have coordinates of the form (x,0,w). The line at infinity has equation w=0, but that’s not really any different from y=0. If I try not to think about where it is, and just focus on the coordinates, then the line at infinity is not different from any other line.

If we want to, we can think of this like magically travelling to stand on the line at infinity, and from that perspective it looks just like any other line. Interesting things happen when you shift perspective this way. For example, consider two parallel lines and a transversal: if you stand on the line at infinity, you can see the point at infinity involved, and the three lines become just an ordinary triangle. We’re going to talk about what happens when you do this to conics.

Conic (sections)

Conics began life millenia ago as literal sections of a cone. You took a circular cone, including both a top nappe and a bottom nappe, and you sliced it with a plane at various angles to produce various results – parabolas, ellipses, hyperbolas. If you arranged it just right, you got a point, or a single line or a pair of lines. These are often called the “degenerate conic sections”.

Later, when coordinate geometry became a thing, it turned out that the conic sections could be described as the set of points whose coordinates satisfy a quadratic equation. Take a quadratic equation in two variables like x² – 3xy + y² – x + 4y – 7 = 0, and find all the points that work when you sub their coordinates into the equation and this give you a conic section. Very nice.

There are a few extra possibilities that appear from our equations that aren’t covered by our cone: a pair of parallel lines, and an empty set where no points at all satisfy the equation. To include these, we’ll drop the word “section” from the name, so that they’re still objects related to cones in some roundabout way, but not necessarily a section of a cone.

Quadratic equations in homogeneous coordinates

When we introduced a third coordinate to our points, and so introduced a third variable to our lines, we simply attached our new variable to the constant term in our line equation. Like this: 3x – 7y + 5 = 0 becomes 3x – 7y + 5w = 0. But it’s not so simple with quadratic equations. In fact, it’s just the slightest bit dodgy to simply attach a w to the constant term, because the x and y in the first equation aren’t really the same as the x and y in the second equation if w isn’t 1. A more rigorous approach would be to remember how to convert from three coordinates back to two like this: (x,y,w) becomes (x/w, y/w). So that “x” in the first equation is actually x/w, and the “y” is actually y/w. Then 3x – 7y + 5 = 0 becomes 3(x/w) – 7(y/w) + 5 =0, and multiplying by w gives 3x – 7y + 5w =0.

(Someone is sure to notice the slight dodginess here too: I’ve still used x and y to mean different things before and after the change. My geometry lecturer back in the distant past switched to capital letters to mark the change, but I think we’ll cope with just a little bit of notational sloppiness for the ease of explanation. I hope so anyway.)

This is how we’re going to deal with quadratic equations. That is, we’ll replace the x’s with x/w and the y’s with y/w:
x² – 3xy + y² – x + 4y – 7 = 0
(x/w)² – 3(x/w)(y/w) + (y/w)² – (x/w) + 4(y/w) – 7 = 0
x²/w² – 3xy/w² + y²/w² – x/w + 4y/w – 7 = 0
x² – 3xy + y² – xw + 4yw – 7w² = 0
(To get to this last line, we multiplied by w².)

So the linear terms get a w attached to them, and the constant term gets a w². This has the effect of meaning that in every term there are two variables multiplied together, either two different ones like xy or yw, or two of the same like x² or w². All the terms are the same in some sense now — homogeneous, as it were.

How conics meet lines

We’re now ready to figure out how our conics meet the line at infinity. Well, nearly. First it’s probably a good idea to know how conics meet ordinary lines! Drawing some pictures, I can convince myself that a line either misses a conic entirely, meets it in exactly one point, meets it in exactly two points, or (in the case of some degenerate conics) is actually part of the conic itself.

The algebra matches the drawings (as it ought to). For example, take a line equation in our old coordinates like 2x + 3y – 7 = 0 and a quadratic equation in our old coordinates like x² – 3xy + y² – x + 4y – 7 = 0. You can rearrange the line equation to get y in terms of x, and sub that into the quadratic equation to get a quadratic equation in terms of just x. And a quadratic equation has zero, one or two solutions, corresponding to zero, one or two points where the line meets the conic. The only other possibility is when everything cancels out in which case every x is a solution and the whole line is part of the conic.

There’s some terminology to name the different kinds of line with respect to a conic. If the line misses the conic entirely, we’ll say “the line is external to the conic”, or say “the line is an external line”. If the line meets the conic in exactly one point, we’ll say “the line is tangent to the conic”, or say “the line is a tangent”. And if the line meets the conic in two points, we’ll say “the line is secant to the conic”, or say “the line is a secant”. (And if the line is actually part of the conic we’ll say “the line is a generator line of the conic”, but that’s not going to come up for our line at infinity.)

A story about me: My PhD thesis was mostly concerned with tangents, secants, generators and external lines of conics and their higher-dimensional friends and relatives, so these words bring back all sorts of memories for me. In particular, I remember giving my second-year thesis seminar the title “David Butler and the Chamber of Secants”. (The first-year seminar was called “David Butler and the Philosopher’s Cone”, by the way.)

How conics meet the line at infinity

Ok, now we’re ready to find how our conics meet the line at infinity. Let’s try the conic I’ve been using as an example so far. Its equation is x² – 3xy + y² – x + 4y – 7 = 0, and in homogeneous coordinates this is x² – 3xy + y² – xw + 4yw – 7w² = 0. The line at infinity has equation w=0, so let’s sub this in to get x² – 3xy + y² = 0. If we treat x as a constant, this is a quadratic equation in y, so we could complete the square/use the quadratic formula to get y = [3x ± √(9x² – 4x²)]/2. Note the x²‘s under the root sign which mean we can pull it out, and indeed pull the x’s right out to get y = x [3 ± √5]/2. So we have two points at infinity of the form (x,x[3 ± √5]/2,0). We can divide by the x to get (1,[3 ± √5]/2,0), which looks a bit nicer. And actually, this means that the points at infinity on this conic are the same as the points at infinity on the lines with slope [3 ± √5]/2.

This is all well and good, but it’s not particularly edifying is it? No, not really. Perhaps it might be best to do this trick with our standard versions of the three non-degenerate conics. That might be a bit more illustrative. I’ll do them in this order: hyperbola, parabola, ellipse.

Hyperbola

Your standard form of a hyperbola equation looks like this (x/a)² – (y/b)² = 1. In homogeneous coordinates this is (x/a)² – (y/b)² = w². When we sub in w=0 we get
(x/a)² – (y/b)² = 0
(x/a)² = (y/b)²
±x/a = y/b
±xb/a = y
So our hyperbola meets the line at infinity at (x, ±xb/a,0) = (1, ±b/a,0). That’s two points, so the line at infinity is secant to our hyperbola. Interestingly, those are the two points at infinity on the lines with slope ±b/a. If you happen to know a lot about hyperbolas, these happen to be the slopes of its two asymptotes!

Actually, let’s just check that shall we? The two asymptotes of the hyperbola with equation (x/a)² – (y/b)² = 1 have equations y = (b/a)x and y= -(b/a)x. In homogeneous coordinates, we have (x/a)² – (y/b)² = w², y = (b/a)x and y= -(b/a)x. Subbing y=(b/a)x into the conic’s equation gives
(x/a)² – (y/b)² = w²
(x/a)² – ((b/a)x/b)² = w²
(x/a)² – (x/a)² = w²
0 = w²
w=0
So this asymptote meets the hyperbola at a point on the line at infinity. I suppose this makes sense — the conic gets closer and closer to the asymptote, so it only makes sense that they actually meet at infinity.

It’s interesting to note that the asymptote meets the conic in exactly one point (x, (b/a)x,0) = (1, b/a,0). That makes it a tangent. Also, even though the two branches of the hyperbola both approach the asymptote, there is only one point where they meet. That does make sense — way back in the first post, we figured out that the point at infinity on any line is both behind and in front of us. If we imagine standing on the line at infinity, you’ll be able to see this point where the conic meets the asymptote. One branch of the hyperbola meets this point on one side, and the other branch meets it on the other side. So here at infinity, the hyperbola links up to itself, like an ellipse.

Parabola

Your standard form of a parabola equation looks like this: y=ax². In homogeneous coordinates, this is yw = ax². When we sub in w=0 we get 0=ax², so x=0. This gives us one point (0,y,0)=(0,1,0). So the line at infinity meets our parabola in one point — it’s a tangent. Interestingly, this is the point at infinity on the y-axis, which is the axis of symmetry of the parabola.

Let’s check this with another parabola. What about x = ay²? In homogeneous coordinates, this is xy = ay². When we sub in w=0 we get 0 = ay², so y=0. This gives us one point (x,0,0)=(1,0,0). So again we get one point, this time the point at infinity on the x-axis, which is again the axis of symmetry of the parabola.

Admittedly, this is only two examples, but I’ll trust for a moment that it works. If I do trust it, it seems if I stand on the line at infinity, it’s tangent to the parabola, and the parabola links up to itself at that spot, like an ellipse.

Ellipse

Your standard form of an ellipse equation looks like this: (x/a)² + (y/b)² = 1. In homogeneous coordinates this is (x/a)² + (y/b)² = w². When we sub in w=0 we get (x/a)² + (y/b)² = 0. Since we can’t have both x and y being zero, this means (x/a)² + (y/b)² has to be a strictly positive number, so we have no solutions to this equation. In short, our ellipse doesn’t meet the line at infinity at all. If you like, the line at infinity is external to our ellipse. I suppose this makes sense — we can see the whole ellipse just fine so it shouldn’t meet the line at infinity.

Only one conic

So it turns out that there is actually only one kind of nondegenerate conic. Hyperbolas and parabolas are really just ellipses that meet the line at infinity in different ways. A hyperbola is an ellipse that has the line at infinity as a secant, and a parabola is an ellipse that has the line at infinity as a tangent. I often imagine it as an ellipse that is stretched out further and further, until it meets the line at infinity to become a parabola, and then crosses the line at infinity to come in on the other side as the opposite branch of the hyperbola.

And I think that might be enough. Except for one parting shot — if we have a random quadratic equation, we have a good chance of knowing what kind of conic it is by seeing how it meets the line at infinity. Our original conic with equation x² – 3xy + y² – x + 4y – 7 = 0 met the line at infinity in two points, so I can be pretty sure it’s a hyperbola. It is possible it’s actually a pair of lines, I suppose — a pair of lines also meets the line at infinity in two points. There is actually a way of telling the difference, which involves making a matrix and calculating its determinant. But that’s another story and shall be told at another time.

14 July, 2016