Math with Art Supplies

The Center of a Circle

After teaching this in a number of classes, I hear that I’ve been getting credit for this method of finding the center of a circle. Just setting the record straight: thanks goes to Greek mathematician Thales of Miletus, who lived just under 3000 years ago.

This is for anyone who has ever used a lid to make a circle, then is left wondering where exactly is the center. What makes this method so accessible is that we all have the essential tools: a pencil, a straight edge, and a piece of paper that has corners.

The surprise is that if you touch the perimeter of a circle with the regular corner of a piece of paper (which is typically 90 degrees, a right angle), draw in the edges of the paper to the point where they intersect the circle,…

…then connect those intersecting points with each other, you end up with a line that is not only the longest side of a triangle, but it’s also the diameter of the circle. Who would have thought that this would be true? Well, Miletus did.

What good is a diameter for finding the center of the circle? Not a whole lot of good, actually. But what if you repeat all of the steps above, but, second time around, start out touching a different part of your circle with the corner of your paper?

Look! The diameters intersect, and that’s the center of the circle.

Just to be sure, there’s no reason not to make a whole lot more of these right triangles.

Great. Now what?

Do whatever you want. That part is up to you.

If you want to give me credit, instead of calling this Thales Circle Theorem, feel free to call it Paula’s way of finding the center of a circle. I’m good with that.

polyhedra · Solids

Polyhedra are like Tribbles

Polyhedrons in my office

“,,,,make a couple more. Decorate your home. Show your friends. Impress your neighbors. Start a club. Make lots more. Pile them high on every horizontal surface you own. Put them in large boxes and carry them with you every time you move. Make even more. They’re like tribbles…” ~George Hart

I’ve been sniffing out polyhedra, aka three-dimensional solid shapes, for some time. The more time I spend with them, the more enchanting they seem to be. There is something so satisfying about seeing what a shape look like in the real world. Pictures, even animated ones, just aren’t good enough.

It’s true what mathematician George Hart says about these. I can imagine his own workspace is busting with polyhedra. I’m trying to space mine apart in different rooms, so as not to give the impression of an invasion. This strategy, predictably, is failing.

It doesn’t help that I’m teaching classes in that focus on these solid forms, as I feel duty bound to make more and more and more. It’s so much fun.

Polyhedrons in the bookcase

Even so, I’ve been enviously looking at my friend Jane’s creations lately. She makes stunning decorations on cookies. I bet she doesn’t have a problem with storage.

I could just stop making them, but that doesn’t seem to be happening. At the moment I have it in my mind to make all thirteen of the Archimedean solids, which may not sound like much, until I tell you that what I’m really interested in are the pieces that can be added back on to make the Archimedean solids into a Platonic solids. I ‘don’t know how far I will get into this project, but it seems like, as I move through the list of the 13 Archimedean Solids that a great number of pieces will debut in the rooms of my house.

Polyhedrons in progress, on my armchair

This Archimedean Solids project has been a long time coming, as I pretty much have been mostly interested in trying to ignore them. I think I knew if I turned my attention towards them that I’d be hooked. I am

Those little edges that are sliced off are the part that I’m taking delight in. For some reason my interest in this part reminds me of something my dad once said to me: after many times of showing him the places I was traveling to he wondered, in the way that fathers do, why is it that I always seemed to be going to the inserts of the maps. Like, there would be North Carolina or West Virginia, or Texas, and we’d have to find that little box that shows the extra part of the state.

Polyhedrons on my desk, a cube (Platonic Solid) and a truncated cube (Archimedean Solid)

An example of what I mean about the parts that come off one solid to make another: above, those delineated corners come off to transform the cube into a truncated cube. Now, guess what shape those corners make if you pack them together. Are you thinking that they make another small cube? That’s what I thought, but I was wrong. Had to make them to believe it: they become an octahedron, which is a shape that’s like two square based pyramids stuck together.

Polyhedral Kindling

With all these Polyhedrons taking over, I try to be extra vigilant about not holding onto some constructions that have some sort of fatal flaw, even if that flaw could be hidden by turning it just so. It’s always sad to see one in with less royal scraps, but it’s better that it keeps me warm instead of possibly, one day, taking over my world.


The Trident Closure

As the first day of teaching Zhen Xian Bao in Depth approaches, I am in constant planning mode. I scrutinize very clever bit of paper engineering that I come across. My husband has recently taken to buying packs of Trident gum, which are packaged in a variety of ways. The pack he brought in last night caught my eye, so I’m recreating it as a cover/closure that I may or may not use in a class.

Tools needed, with paper recommendations

The fact that Susan Joy Share, who co-instructs the ZXBinDepth class with me, knows and creates such a wealth of closure solutions makes it unnecessary for me to work out any more. Last year Susan showed our classes some examples of what people call Nag Hammadi closures, named for the methods used on some ancient books found in Upper Egypt. She also developed some a gorgeous crocheted closure that seems to grow out of the cover it is on, as well as demonstrating ways to use magnets to make our folded forms shut with a satisfying click.

Folder with Closure for a mini-Chinese Thread Book

With such a treasure chest of methods that are already part of our plans, adding the Trident Gum packing option seems dubious. Still, it’s so darn sweet that I’m going to park it here.

Here’s the video of how to make this:

Zoom Meeting

Folding for Ukraine

Folding for Ukraine

Here’s an event that deserves its own post. I very much want to spread the word about this opportunity to show support to Ukraine while at the same time spending time with some incredibly special people.

This Sunday, January 29, 2023 the organization Folding Didactics is hosting a day-long conference, Folding for Ukraine, They are asking for a mere donation of  2 $/€/£, which will go towards the work they do to show solidarity with the Ukrainians struggle for freedom.

Paul Jackson

The cause is stellar, and so is the line-up. Three of my heroes in one zoom event! Paul Jackson, whose books I’m been devouring for years, will be there.

An Origami Snake that I learned watching Dr Lizzie Burns during a live folding session in March 2021

Dr. Lizzie Burns, whose approach to teaching paper folding completely captivated me when I found her during the early days of Covid lock down days, will also be presenting.

Joan Sallas

Joan Sallas, a highly skilled folder, passionate scholar, and delightful instructor, provides the final workshop of the day. Joan recently showed his amazing collection of Zhen Xian Bao to a group of Ukrainian teachers, which I was lucky enough to have joined through zoom. Joan lives in Spain, gave his presentation in French, which was then translated into in the language of Ukraine. It was amazing to see many of the Zhen Xian Bao in his collection. I have been a fan of Joan’s research and instruction for as long as I’ve known about him. He will be talking about methods for teaching folding, which is something I absolutely don’t want to miss.

Nick Robinson, Lee Armstrong, and Micheal LaFosse will be presenting earlier in the day. I don’t these three gentlemen, but am looking forward to getting to know them.

The final event of the day will be Ukrainian teachers talking briefly about their work and their experiences in using origami during the war.

As I’m waking up here in New York at 7am Sunday morning, the folding event will just be heading into their 3 pm afternoon break. When they resume at 4 pm EEST/ UTC/ GMT+2 time it will be 8am here. Fortunately, Folding Didactics will make the recordings available to registrants. Needless to say, I’ve registered. Hope you check it out, too.

Folding For Ukraine