BLOGS WEBSITE

It’s been four years since I came up with the idea of iplanes as a way to organise the complex points on a graph, and in the intervening time I have thought about them on and off. For some reason right now I am thinking about them a lot, and I thought I would write down some of what I am thinking. It will take several blog posts, and they are likely to be some weeks apart, so bear with me.

A recap of the iplane 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 (p,q) are all of the form (p+si,q+ti). That is, the real parts of the two coordinates are (p, q). 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 call this plane of complex points the iplane at (p,q). Every complex point is in one of these iplanes. 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).

iplane-at-3-2

I imagine the iplanes as transparent sheets attached at each real point, that are usually rolled up like an umbrella, but can be flattened out to sit on top of the ordinary real plane when you need to. And yes I imagine them as transparent red or pink, because that’s the colour cellophane I had in my original model.

all-iplanes
iplane-at-3-2-unfold

The points on a complex line

Last time, I did a thorough investigation into the points on a complex line, stepping through the points on a real line, a line with real slope but non-real intercept, and a line with non-real slope. But I’ve always been a little unsatisfied with my treatment of it. I really wanted to have a unified approach to all three types of lines, since they are all just complex lines. I was sure there was a way of dealing with them that could allow me to just take a complex line with a standard equation and know what it should look like.

Well there is a way and I’ve done it. I am very proud of it, and not least of the things making me proud is a GeoGebra page that allows you to choose any complex line and then see what points are on the line in the iplane attached to a moving point, but with all kinds of complex lines together in one GeoGebra tool, rather than separate like last time. But we’ll get to that soon…

First, the calculations.

Consider a line L with equation (a+αi)x+(b+βi)y=(c+γi), for real numbers a, α, b, β, c, γ with a, α, b, β not all zero. This is the most general form of a complex line. Given a real point (p,q), I am going to find all the complex points in the iplane attached at (p,q) that are on the line L. I will suppose that the point (p+si,q+ti) is on L and try to find the imaginary parts of its coordinates (s and t).

Since (p+si, q+ti) is on the line L, it satisfies the equation, and so:

(a+αi)(p+si)+(b+βi)(q+ti)=(c+γi)
ap + asi + αpi + αsi² + bq + bti + βqi + βti² = c+γi
ap – αs + bq – βt + (as+αp+bt+βq)i = c+γi

Equating real and imaginary parts, I get two equations:

1: ap – αs + bq – βt = c
2: as+αp+bt+βq = γ

Which I can rearrange into:

1: αs  + βt = ap + bq – c
2: as + bt = γ – αp – βq

Whatever points (p+si,q+ti) inside the iplane at (p,q) are on the line L, they must have their s and t’s satisfy these two equations.

But look closer at them! These two equations are the equations of lines! So whatever points in the iplane lie on both of these lines are the points of the original complex line L in this iplane. Two lines meet in exactly one point, which means there is exactly one point in the iplane on the complex line L!

They are not just any lines either: the coefficients are chosen from the real and imaginary parts of the original line equation. Equation 1 has s and t coefficients which are the imaginary parts of the original line’s coefficients, while Equation 2 has s and t coefficients which are the real parts of the original line’s coefficients. That means these two lines inside the iplane are parallel to the “real part” and “imaginary part” of the original line equation, with equations ax+by=c and αx+βy=γ respectively.

And those two equations appear again in the equations inside the iplane at (p,q).  The constant term in Equation 1 is ap+bq-c, which you would get if you substituted the point (p,q) into the real part of the original equation. The constant term in Equation 2 is γ-αp-βq, which you would get if you substituted (p,q) into the imaginary part of the original equation. If (p,q) satisfies both of these equations, then Equation 1 and 2 both have constant 0, and so have s=0, t=0 as a solution. But if s=0 and t=0, that means the point in the iplane is (p,q), which is a real point! That is, there is one real point on the complex line L with equation (a+αi)x+(b+βi)y=(c+γi), and this one real point is the intersection of the two real lines with equations ax+by=c  and αx+βy=γ. I think that is so cool!

Technicalities

Of course, I have glossed over a few technical details. What if the two lines with equations αs+βt=ap+bq-c and as+bt=γ-αp-βq are parallel? Then there would be no intersection and so no points of the line L at all in this iplane. When would this happen? This would happen when the coefficients α and β were the same multiple of a and b, so α=ka and β=kb for some k. And then the original equation for L would have been

(a+kai)x + (b+kbi)y = c+γi
a(1+ki)x+b(1+ki)y=c+γi
ax+by=(c+γi)/(1+ki)

So the line L can be rewritten to have real coefficients for x and y, which means it has real slope! So a line with real slope will have many iplanes with no points of the line at all. Which ones will have actual points? Well, two parallel lines will share points (indeed all their points) if they are actually the same line after all. This would happen when as well as α=ka and β=kb, we also have the constant terms being related in the same way:

ap + bq – c = k(γ – αp – βq)
ap + bq – c = kγ – kαp – kβq
ap + bq – c = kγ – kkap – kkbq
ap+k²ap + bq + k²bq = kγ+c
ap(1+k²) + bq(1+k²) = kγ+c
ap + bq = (kγ+c)/(1+k²)

This is the equation of a straight line parallel to the one with equation ax+by=c, which is the real part of the original line L’s equation! So either an iplane will have no points of L at all, or it will have a whole line of points of L, whenever the point (p,q) is on a specific line parallel to the real part of the original line.

If the line L itself was a real line all along, then this could only happen if (c+γi)/(1+ki) was real, so γ would have to be kc as well, and the original line equation becomes ax+by=c. There will be points in the iplane at (p,q) when ap+bq = (kγ+c)/(1+k²) = (kkc+c)/(1+k²) = (k²+1)c/(1+k²) = c. This is the actual real line again! So a real line has points only in those iplanes attached to the real line itself, and in those iplanes, the points form a line parallel to the real line.

There is one final technical detail. What if both a and b are zero, or both α and β are zero? If both α and β are zero, then the two equations inside the iplane become

1: 0 = ap + bq – c
2: as+bt=γ.

If ap+bq-c is not zero, then Equation 1 has no solutions and there will be no points of the original line L in this iplane; if ap+bq-c is zero, then Equation 1 is 0=0, which is always true, and so any point satisfying Equation 2 is a point in this iplane. But wait, the points satisfying ap+bq-c=0 are exactly the points on the real line with equation ax+by=c. And since α and β are zero the original equation was ax+by=c+γi anyway. So this matches with what we found before: a line of real slope has points only in the iplanes attached to a real line parallel to the real part of the original line. In this case, not just parallel but equal to the real part of the original line!

The same thing happens when both a and b are zero. Then the two equations inside the iplane become

1: αs + βt = -c
2: 0 = γ – αp – βq.

And now there are no points of L in the iplane unless αp+βq=γ , which is still the equation of a real line. Indeed, the original equation is (αi)x+(βi)y=c+γi, which is the same as αx+βy=-ci+γ, so the line with equation αp+βq=γ is actually the real part of one version of the original line’s equation.

Conclusion

So after all this, we find that the complex line L with equation (a+αi)x+(b+βi)y=(c+γi) behaves like this in the iplanes:

  1. The points (p+si,q+ti) of the complex line L in the iplane attached at (p,q) satisfy the two equations αs+βt=ap+bq-c and as+bt=γ-αp-βq.
  2. If the complex line L’s equation cannot be turned into one with real coefficients for both x and y, then every iplane has one point of the L, at the intersection of two lines inside the iplane. There is exactly one real point on the line L, which is the intersection of the real lines with equations ax+by=c and αx+βy=γ.
  3. If the equation can be turned into one with real coefficients for both x and y, then there are no points of the line L in most iplanes. The only iplanes with points on L are those attached to a real line parallel to the real (or imaginary) part of the original line equation.
  4. If the equation can be turned into one with real coefficients for x and y and real constant term, then L is a real line and the only iplanes with points on L are attached to the points on L itself. The points of L in each iplane form a line appearing to lie on top of L itself.

You can investigate all of this in this GeoGebra applet, which allows you to choose coefficients for the equation of L and move the point (p,q) to see the point(s) of L in the iplane attached there.

Thanks for being here for my mathematical scribblings. I worry that it might not make any more sense than the old version, but I feel a deeper sense of satisfaction having tried it both ways (and having one GeoGebra applet with all the types of lines at once).

Posted in Isn't maths cool? | Tagged , ,
Leave a comment

This blog post is about a piece of the MLC learning environment which is very special to me: the date blocks. It’s a set of nine blocks that can be arranged each day to spell out the day of the week, the day number, and the month. I love changing them when I set up the MLC in the morning, so much so that since the face-to-face MLC closed due to COVID-19, I brought them home and have been changing them each morning here in the dining room.

The story of how this object came into the MLC is the reason it is so special to me.

It began with a Christmas countdown calendar that my wife’s brother and his wife gave to us one year. It has two blocks that are able to spell out how many days there are until Christmas. Every year in our house we hang it up on the 1st of December and take turns changing it each day.

A couple of days before Christmas in 2012, I was thinking about how very soon it would be 365 days until Christmas. I knew that even though we only use our countdown calendar for the numbers from 25 to 00, it is possible for it to count down from 32, and I wondered if you could make a set of blocks that counted all the way down from 365.

I asked this question in the online One Hundred Factorial discussion area that we had at the time. Over a couple of weeks of very slow discussion, we discovered you can only count to at most 87 with three blocks if you must have all three on display, and so you need at least four and (possibly even more) blocks to be able to count all the way from 365 to 0.

At this point someone suggested that since we already had the ability to spell 1 to 31 with two blocks, maybe we could use three extra blocks to spell the month. This was an intriguing suggestion. If you’d like to think about it yourself for a bit, then I suggest you look away now, because there are about to be spoilers…

If you look at the first three letters of all twelve months, there are 19 letters: JANFEBMRPYULGSOCTVD. This would seem to say it’s not possible to spell all the three-letter months with three blocks because three cubes only have 18 faces.

However, the number blocks that inspired all of this are only able to produce all the numbers they do by allowing the 6 to also represent a 9 when you turn it upside-down. So we thought maybe there are letters in the list that could be other letters if you turn them the other way. The only candidates we could see were U and C if we drew them in exactly the right way.

Unfortunately, even with the C and U as the same letter, it still wasn’t possible. Since we have to spell both JAN and JUN, that means that A and U have to be on the same block. But since we also have to spell AUG, that means that A and U have to be on different blocks. They can’t be on both the same block and different blocks, which means it’s not possible to do it with three blocks. (This right here is one of my favourite proofs by contradiction of all time.)

Or is it really impossible after all? We couldn’t see any letters that could be turned into each other, but we were using capital letters. I suggested that perhaps we could use small letters instead. Then the u could be turned upside-down to be an n, which means it might be possible for a and u to be both on the same block and different blocks, if the u from the other block was actually an upside-down n. We still need to have both u and n for this of course, so unfortunately that’s still 19 letters to fit onto 18 faces. The letters p and d to the rescue! They can be rotated into each other, and they don’t have to both appear in the same month abbreviation!

Now you know why I used lower case letters for my date blocks: because it’s not possible with upper case letters!

Of course, you still have to figure out exactly how to put the letters onto the blocks, and that I was able to do on the 14th of January 2013. This is a photo I took at the time.

But I couldn’t stop there. The days of the week also have three-letter abbreviations, and so I made a set of three blocks for those too. These were much quicker to figure out because there are only seven days after all!

This left the day number, which I knew already could be done with two blocks, but it seemed weird to have only two blocks for that when the weekday and the month had three each. I decided that since I wanted to use three blocks, I might have the leeway to have blank faces so I could spell the one-digit numbers without a leading 0, and so that there weren’t redundant faces, I decided to try to include the “st”, “nd”, “rd”, “th” suffixes too. I did in fact figure out how to do this, though I am not at all sure it’s the most efficient solution.

And so after a day of avoiding other work by playing with cubes, I came up with a solution on the 15th of January 2020. I chose to display the 1st of January so that I could make it clear there were blank faces and more than just “th”.

I put these paper blocks on display in the MLC next to the sign-in sheet box (yes we had a paper sign-in system in 2013), and I changed them every day through Summer School. When first semester arrived with a crop of new students, I decided it was high time for a sturdier more permanent-looking set. This set is made of cardboard, which was painted in dark paint, pasted with printed letters, then painted over with PVA glue as a lacquer.

This would be the end of the story except for one extraordinary epilogue. One of our regular students had observed me changing the blocks every day, and had noticed that while I could stack them on top of each other to make the date visible, they were a little precarious. So he took measurements and made a case for them that they could fit into, to be properly on display. He presented it to me on 22nd April 2012. The most impressive part is the handle at the top, which is an MLC and Writing Centre branded pencil. The MLC and Writing Centre shared a room when we moved into the Hub Central building in 2011 and even though they moved into their own room just around the corner in early 2012, we used these pencils for some years later.

So that’s the story of the MLC date blocks. They are special to me because they grew out of a wondering, that became a puzzle at One Hundred Factorial, that inspired a new puzzle following our MO of “the goal is not the goal”, and the solution has some really lovely reasoning and problem-solving. They remind me of all this wonderful mathematical thought, but even more than this, they are also connection to a student who loves the MLC environment so much he would build something for us.

Not many things hold so many special memories, and a calendar seems an appropriate sort of thing to do so.

Posted in One Hundred Factorial, Other MLC stuff
Leave a comment

The weekly puzzle session that I run at the University of Adelaide is called One Hundred Factorial. In the middle of the night, I suddenly realised that I have never written about why it is called One Hundred Factorial, and so here is the story.

The very beginning

Once upon a time I was a PhD student in the School of Mathematical Sciences at the University of Adelaide. Sometime during the third year of my PhD program (2007), I was asked to give a talk to the first year undergraduate students as part of an evening event where the goal was to hopefully convince them to keep studying maths at a higher level next year. I titled my talk “How to Tell If You Are a Mathematician”. I don’t remember any of the things I spoke about, except for one thing. Before I started talking, I put a puzzle up on the document camera. I did not mention the puzzle in any way or look at the screen at all. I just did my little talk as if it wasn’t there. But right at the end of my talk I said this:

The final and truest way to tell that you are a mathematician is that you haven’t been listening to any of what I just said, and instead have been trying to solve this puzzle.

Cue guilty looks and nervous laughter from all of the academic staff in the audience, which successfully proved my point. Anyway it worked. Several students came up to me to talk about the puzzle, and I was able to direct them to lecturers who could talk to them about their study options. Yay for puzzles, right?!

This was the puzzle I used so neatly to make my point about the mathematician’s mind:

The number 100! (pronounced “one hundred factorial”) is the number you get when you multiply all the whole numbers from 1 to 100.
That is, 100! = 1×2×3×…×99×100.
When this number is calculated and written out in full, how many zeros are on the end?

I don’t remember where I got the puzzle from, but it is a pretty famous one that’s been around for some time. I actually hadn’t even thought through a solution at the time either. I just knew that it mentioned a concept that had been in the first year lectures recently.

The puzzle sessions begin

The other thing that happened that night was that a group of students and staff stood at the blackboard in the School of Maths tea room to nut out a solution to the 100! puzzle. I can’t even remember if we finished it or not, but we did decide that we should get together regularly to solve puzzles together, and a weekly puzzle session was born. At the first session, we started with the 100! problem again, and an extension of it, which is to find out what the last digit is before all those zeros start. Then as the weeks went on, we would do puzzles that I would find and bring to the sessions.

When I finished my PhD in mid-2008 and took up the job in the Maths Learning Centre, I took my little puzzle session with me, and was able to invite more students to come along, and it slowly morphed into a student event more than a staff event, which really pleased me. In fact, a regular at these puzzle sessions for years was that first student who had come up to me after my talk at the first-year event, and he eventually became one of my tutors at the MLC.

The name of the event

Over the years the puzzle session has had many names. We started out calling ourselves “People with Problems”, and then simply “Puzzle Club”. For a while it was called “The Hmm… Sessions” after the sound we made very often while thinking about puzzles. Indeed, there is a reference to the Hmm Sessions inside this very blog. But in 2012 after the website where I was hosting our online discussion was decommissioned, I decided it was time to change the name. I was also starting to think about moving the sessions out of the MLC itself and into a public space, and to match with this move I wanted a new name. I thought long and hard, and decided to name it after the first puzzle we ever did, the puzzle that first inspired staff and students to talk and think about maths together, the puzzle that helped students decide they really were mathematicians after all.

The legacy

So the regular puzzle session of the MLC became One Hundred Factorial at the end of 2012, and here we are in 2020 still going, so that now it’s been One Hundred Factorial longer than it’s been any other name. It’s been my testing-ground for new puzzles and games and teaching ideas, a place where I have made friends and welcomed people from around the country and the world. And it has become a glowing island of mathematical play in the middle of the stressful university life, and indeed the middle of a stressful life generally. In recent weeks it is a glowing island of community in a world of pandemic-induced isolation.

One Hundred Factorial reminds us that there is always something joyful to think about if you are looking for it, and that it’s okay to pause and ignore your responsibilities for a while to think about it, and that doing this with people is a source of shared joy. I hope the puzzle and the event can keep reminding us of that for a long time yet.

Posted in One Hundred Factorial | Tagged ,
Leave a comment

I don’t like BODMAS/BEDMAS/PEMDAS/GEMS/GEMA and all of the variations on this theme. I much prefer to use something else, which I have this week decided to call “The Operation Tower”.

In case you haven’t heard of BODMAS/BEDMAS/PEMDAS/GEMS/GEMA, then you should know they are various acronyms designed to help students remember the order of operations that mathematics […]

Posted in How people learn (or don't), Thoughts about maths thinking | Tagged ,
Leave a comment

This blog post is about a game I invented this week, and the game is AWESOME, if I do say myself.
DIGIT DISGUISES: A game of algebraic deduction
Players:

This game is designed for two players, or two teams.

Setting up:

Each player/team has two grids with the letters A to J, one labelled MINE and one labelled THEIRS, like the […]

Posted in Isn't maths cool?, One Hundred Factorial, Thoughts about maths thinking
4 Comments

Context fatigue is a particular kind of mental exhaustion that happens after having to make sense of multiple different contexts that maths/statistics is embedded in. I feel it regularly, but I feel it most strongly when I have spent a day helping medical students critically analyse the statistics presented in published journal articles.
The problem with maths […]

Posted in Being a good teacher, Thoughts about maths thinking | Tagged ,
Leave a comment

I am often saying how important it is to listen to students, and that I am fascinated by student thoughts and feelings. When students say I am a good teacher my usual response is to say it’s because I have spent the last eleven years in a situation where I get to listen to lots […]

Posted in Being a good teacher | Tagged ,
Leave a comment

Three years ago, my university’s Student Engagement Community of Practice collectively wrote a series of blog posts about various aspects of student engagement. I thought I would reproduce my blog post here, since it is still as relevant today as then.
There is a lot that staff can do to engage students in the university […]

Posted in Being a good teacher | Tagged
Leave a comment

I firmly believe that all students deserve to play with mathematical ideas, and that extension is not just for the fast or “gifted” students. I also believe that you don’t necessarily need specially designed extension activities to do exploration — a simple “what if” question can easily launch a standard textbook exercise into an exploration.
This […]

Posted in Being a good teacher, Thoughts about maths thinking | Tagged ,
Leave a comment

Once upon a time, I decided I would be vulnerable on Twitter. As part of that, when someone posted a puzzle that I was interested in, I decided that I would not wait until I had a complete answer to a problem before I responded, but instead I would tweet out my partial thinking. If […]

Posted in Thoughts about maths thinking | Tagged ,
Leave a comment