Being who I am, I began with geometry. A disk takes 6 or 7 increases around because when you increase the radius of a circle by 1 unit (i.e. by one round) the perimeter increases by 2π units, 6.28ish. We have to fudge a little, of course, since an sc doesn’t add exactly the same amount to circumference as to radius and we can only increase by whole stitches, but it works out; we are able to make disks.

For a polygon, there are two distances that could play the role of the circle’s radius: center to corner (radius), and center to edge midpoint (apothem). We have formulas that tell you how much the perimeter increases when the radius or apothem increases by 1, depending only on the kind of polygon you’re expanding.

Shockingly, I’ve decided not to go into the algebra here; you can read all about it Math Open Reference. My previous knowledge says you need 8 extra stitches for a square, and that number should be larger for fewer sides and smaller for more sides (you need more stitches to get around pointier corners). Those both matched the apothem calculation and not the radius calculation.

The apothem numbers leave a lot to be worked out: how to round, what to do when the increases aren’t a multiple of the number of sides, and whether an octagon could even be made when it called for fewer increases per round than corners. I made all six polygons more or less successfully, but they broke out into half easier, half harder.

The easy polygons were the triangle, square, and heptagon.

Triangle: This didn’t go how I expected – I thought I would need to round up to 12 extra stitches per round, but I actually dropped down to 9. I started with 6 sc in a magic ring, and every corner got 4sc. Increases made into previous increases went into the third of the four sc.

Square: As I said, I already knew to put 3sc into the corners to make a square. I started with 6 sc, increased around, and then started making concentrated increases for corners. Increases made into previous increases were made into the middle sc.

Heptagon: Since for me, seven increases is appropriate for making a flat disk, the heptagon was straightforward. YMMV. I started with seven stitches, increased around, and then increased in the second stitch of each previous increase. To improve the point of the corners, in the last round I made 3sc into the second stitch of every previous round increase.

Pentagon, hexagon, and octagon were more difficult, but they did work reasonably well.

Pentagon: The pentagon formula called for 7.3 new stitches per round. Since five 2sc increases would add 5 and five 3sc increases would add 10, I alternated between them: start with 5 sc in a magic ring and make 3sc into each of them. Next round, put 2sc into the center of each 3sc increase; round after that, put 3sc into the second of each 2sc increase. Continue alternating, ending on a 3sc round. I did attempt mixing 2sc and 3sc increases within individual rounds, but it was a mess to keep the side lengths equal.

Hexagon: Like the pentagon, I used a combination of 2sc and 3sc increase rounds. The hexagon’s apothem number was lower and the number of increases per round higher (6 or 12) so I made two 2sc increase rounds for every one 3sc increase round. It perhaps would be even better to make three 2sc rounds per 3sc round, but I worried about maintaining the flatness of the piece. Start with 6 sc in a magic ring, make 3sc into each of them, and then make two rounds of 2sc increasing and one of 3sc. Put your increases into the second stitch of a 2sc predecessor or the middle stitch of a 3sc predecessor, and for best results end on a 3sc round.

Octagon: How can one even make an octagon if even one increase per corner leads to too many stitches around for the piece to stay flat? I suspect the best answer is to make a disk large enough to naturally hit a multiple of 8 stitches around and then do something like (sc, hdc, sc) in each corner on the last round. I wanted to try to stick to the size and methods of the other polygons (though I didn’t quite) and ended up with this: 7 sc in a magic ring; 2sc around; *2sc, sc* around. You’re at 21 stitches. Make a big jump to 32: *2sc, sc* 10 times, 2sc. Last round: sc 2, *(sc, ch, sc), sc 3* 7 times, (sc, ch, sc), sc. The chain in the middle of the last round’s increases gives it a little bit more point without adding even more extra bulk than we already have.

There you have it: all the polygons from 8 sides down rendered in crochet, for your freeform delight. I did these all in spirals and ended with a needle join in the second stitch; the ultimate perimeter would be smoother if you worked in joined rounds.

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However. There is a far easier way if you own a paper trimmer. Start with a sheet of paper (or taped-together sheets of paper) big enough to accommodate your circle, and fold into quarters. The picture below is a sheet of letter paper from which I will cut a circle 8″ across.

The second picture above shows the first cut. The corner where the two folds meet will be the center of the circle; I’ve placed it 4″ from the blade to cut my paper down into a 4″ square. After that I’ll start cutting off corners that show up between the two folded edges, as shown below.

The second picture above is after six cuts, and it’s already looking pretty good. I didn’t keep count but I would estimate it took 20 or so cuts to make the finished circle, shown folded and unfolded below. Larger circles take more cuts.

You could obsess over the smoothness of your circle and even take scissors to it after the paper trimmer has reached its limit, but the one shown is plenty smooth for the purpose of a sewing pattern – the roundness of my seam would not be improved by additional trimming.

Limitations: with my paper trimmer I could make circles 1.5″ across (the metal strip is 3/4″ wide) or anything 2″ across or bigger (the markings start at 1″), but it’s not so easy until about 3″ across. However, for smaller circles there’s usually something I can trace, or at worst, the ruler method doesn’t take as long. Though circles up to 23″ across are possible, the early stages of large circles are difficult because your paper will likely be wider than the opening for the blade. I recommend folding your paper a third time, into a not-quite-triangle. That extra fold can lead to inaccurate cutting, so trim away excess paper as though you were making a slightly larger circle. Three cuts (two sides and the center) should be plenty to remove the paper that’s in your way, so you can unfold to quarters and proceed with the circle making.

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In 2012 I cut out triangles of paper to glue together into a dodecahexaflexagon (documented in a post about a smaller flexagon). I also found instructions: scanned typed instructions from David Pleacher, and instructions incorporating triangle orientation from Kathryn Huxtable.

A dodecahexaflexagon is a 12-faced (the dodeca-, as you will know if you’ve read The Phantom Tollbooth, or been a long-time reader of this blog), 6-sided (hexa-) flexagon; each face is made from 6 equilateral triangles. I had cut each face from a different scrapbook paper, and I had small squares of white paper to serve as hinges.

In the summer of 2014 I dug out the paper pieces and started gluing them together. I glued one side of the strip together in an evening, but didn’t get back to the other side until now. The second side was quite easy, since on side 1 the faces were scattered around and on side two they were much more orderly.

There was some confusion in the folding and a length of time before I found all 12 faces. I didn’t know the trick! To flex, you’ll pinch the hexagon so that three of the lines between triangles are outward corners and three are inward corners (see photo below). Which edges are inward and which outward will change which face you see next (in some cases you’ll only be able to flex in one configuration). To see all of them, you can simply pinch out the same corner over and over again, only rotating to a neighboring corner if it is impossible to flex the first one. I found hanging on to the same pair of faces with one hand, doing the rest of the work with the other, was the best way to enact that. It is awkwardly thick and I’m glad I spaced the triangles apart a bit with the paper squares.

Each face is connected to at least two additional faces. I haven’t explored thoroughly enough to know whether I found the full set of options, but I made a little map and had each face connected to 2, 4, or 6 others, with complicated interconnection. This lines up with a diagram on Kathryn Huxtable’s general flexagon page, where I also learned about the “pinch one corner repeatedly” method of finding all the faces.

Want more flexagons? Harold McIntosh has an interesting read about the history and theory of flexagons, and Vi Hart’s videos and more (the first of which inspired my flexagon crafting) are all on a hexaflexagon page of puzzles.com. Woolly Thoughts, a bastion of mathematics-inspired crafting, has a page of crochet and knit flexagon cushions.

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Today is tau day, 6/28. In its honor, I have a tau for you. This is made by cutting pi in half rather than doubling it: you need the Big and Little Pi pattern, but then you simply do only select parts of it to make tau.

Here is how to make tau from Big little pi:

• Big little pi top bar: stitch rounds 1-17 (horn through first leg opening). Proceed to stitch rounds 28-33 (making the leg before stitching round 33, as directed).
• To check your counting: there will be 6 rounds of “sc around” after the round that stitches into the chain of the leg opening.
• Big little pi tapered foot: make as instructed, beginning in the center skipped stitch of round 12.

That’s all!

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Big and Little Pi Pattern

You saw a Pi and pi I made for my sister back in late 2012, with the pattern for little Big Pi. That version of big little pi required sewing to shape the curved foot. (The little Big Pi shown also had one leg longer than the other, due to a counting error.)

This version is sewing-free! The pointed foot and “horn” of the top bar curve through a combination of increasing/decreasing and a method I developed to collapse pairs of rounds into single rounds on the inside of the curve. The stitching takes some paying attention, but it isn’t difficult and all stitches are common (very common, in fact: slip stitch and single crochet).

But enough about that. I was fortunate enough to capture some pictures of Pi and pi in their usual haunts…

They love the soothing motion.

I think they’re gambling?

Where they go to relax!

Their individual beauty shots are on the pattern page.

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Less long ago than geometry, I taught calculus. I remember having a bit of trouble with volumes by shells and slicing when I was learning calculus. A curve is rotated around a line to delineate a three-dimensional object, and the goal is to find to volume of the object. For example, a semicircle rotated around the line that joins its tips gives a sphere. Shells and slicing are the two general methods to find the volume. When I taught them as a graduate student I thought it might help to have physical instantiations of some of the regions we might find volumes for, in a form that could come to pieces in either cylindrical shells or sliced disks. Intuition is often helped by seeing some concrete examples in explicit detail. Enter the hardware and craft stores.

The slicing method is the more straightforward of the two; partition the object into disks by slicing through it perpendicular to the axis of rotation. Find the volume of each disk and add them up, using limits and hence integrals to make this precise.

The shell method is what tends to trip people up. Sometimes the slicing method is a no go, and in that case you can peel layers off from around the outside of the volume as though peeling the label off a bottle. These layers are called shells and are cylinders with the axis of rotation through their center. Again you can find the volume of each cylinder and add them up, passing to the limit/integral for precision. Two models are shown below, and the picture heading this entry is a third.

I made these models with nuts, bolts, Sculpy, and aluminum foil, baking the clay with the bolt through the middle. I don’t remember how many layers of foil I used, but I would make it at least 2 for the space needed to dismantle these easily and without risking breakage. Two layers around the bolt and two between each shell. The slices can be spaced apart for baking, so they don’t need foil separators.

It’s been a long time since I taught volumes of revolution, and I think they may have been slightly helpful for some students, but they were not the breakthrough I would have wished. I think that was due partially to how I made them in particular, though. Were I to do it again I would go much larger scale. That would allow me to demonstrate them from the front of the classroom before passing them around, mitigate the problems of inexactness, and permit more layers in the shell case, all of which would help. I might even make the shell models out of fabric, stiffened with interfacing and padded with quilt batting.

It seems likely to me there are some computer animations that attempt to convey the same ideas as these, but I haven’t seen them – there are so many animations I would have made for teaching calculus had I had any idea where to start in creating them. However, for some students, nothing compares to putting their hands on something, so I perhaps putting this idea out in the ether can help someone who’s trying to help such a student.

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When I was in college I tutored a high school student in mathematics for three years. I took it as a personal challenge to create widely varied methods of explanation and practice to get the material not just to stick, but to be understood and retained in a flexible way. When we were doing geometry, one of my attempts was physical instantiations of triangles and quadrilaterals with dowels and rubber tubing. All the individual pieces plus measurements are after the cut!

A rectangle has opposite pairs of parallel, same-length sides and a right angle in every corner. The diagonals bisect each other but are not necessarily perpendicular.

I didn’t get a picture of this, but you can squish the rectangle into a nonrectangular parallelogram, where opposite sides are still parallel and same-length, but the corners are no longer right. The diagonals change length and again, are not necessarily perpendicular, but they still bisect each other.

A kite has two pairs of same-length sides, but they are adjacent, not opposite as in a rectangle. The angles of the corners are determined by the lengths of the sides. The long diagonal bisects the short one (if they are equal they bisect each other), and they are perpendicular.

A trapezoid has one pair of parallel sides, opposite to each other. The remaining sides may be the same length, in which case the trapezoid is symmetric across the midpoint of the parallel sides, or they may not be (a scalene trapezoid, the second picture below).

A rhombus has all four sides the same length. Its diagonals are perpendicular and bisect each other.

A square is all of the above simultaneously: four same-length sides meeting at right angles. The diagonals are the same length as each other and bisect each other at right angles. This square is the same object as the rhombus above.

Before dismantling this all I measured the dowels. Here’s what you need to make your own:

1/4″ rubber tubing and 3/8″ dowels
Rhombus/square: 5″ sides; square diags appx 7″, rhombus diags appx 5 7/8″ and 8 1/16″
Kite: 4″ and 7″ sides; diags appx 5 5/16″, 9 5/16″
Trapezoid: three 4″ sides, one 7″ side; diags appx 6 5/8″
Parallelogram/rectangle: 4″ and 7″ sides; rectangle diagonals appx 8 1/16″; parallelogram diags were lost so measure your own for the degree of slant you’d like

Somehow I did not measure the scalene trapezoid. You could also make a general quadrilateral without any special characteristics.

Triangles are straightforward: right (3-4-5, which could be 4.5-6-7.5 for ease of use if desired), equilateral (4-4-4 or 5-5-5), isoceles (maybe 4-5-5), scalene (4-5-6?). I’m not sure how beneficial they are to the student, however, compared to the quadrilaterals: seeing the deformation of square to general rhombus, rectangle to general parallelogram; comparing a kite and parallelogram with the same set of side lengths; seeing the how diagonals intersect.

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To flex from one face to the next, you fold the long edges away from you and open down the middle. When 1 and 2 are showing, 1 will not open but 2 will reveal 3. When 2 and 3 are showing, 2 will open to 1 and 3 will open to 4. Finally, as you would expect, when 3 and 4 are showing 3 opens to 2 and 4 does not open.

Of course, after making lovely schematics for you, I remembered to do a search to try to find this. It’s a tetra-tetraflexagon, known since at least 1961. I think my instructions still add something, but I’ll put them under a cut.

Meanwhile, I made a new one without numbers. Unfolded it looked as follows:

And here’s another animation!

Start with a rectangular piece of paper, divided (perhaps only mentally) into twelve rectangles on each side such that every rectangle has equal width, and every rectangle has equal height (i.e. the vertical lines are equally spaced and the horizontal lines likewise). My rectangles were 1″ squares in the original flexagon, and 1.5″ squares in the new one. In these diagrams, the lines that don’t meet any other lines are there to indicate the boundaries of squares. All lines that meet other lines are edges of the paper.

The middle two squares are cut away from the rest on three sides, including both longer sides. When the “hinge” of this flap is to the left, the squares belong to faces as follows:

Turning the paper over, side to side so the same long edge is at the top, the squares belong to faces as indicated in the next diagram.

To fold, keep this latter side up and fold the middle flap out to the right.

Fold the leftmost column in once…

…and twice.

Turn the folded paper over and check that you have the correct faces showing.

Finally, fold the last tab in to the center and secure with a piece of tape.

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I used ordinary cotton calico, the kind sold for quilting. I have some doubts that Mod Podge would work terrifically well to glue anything very heavy or thick. I cut pieces to wrap around two binder clips and cover a promotional magnet, brushed Podge on the surface of the item, pressed the fabric onto it (this required trimming to fit in all cases, more so for the binder clips), and brushed more Podge onto the surface of the fabric.

As you can see, it worked beautifully, with the caveat that the color of the base shows through the fabric a bit. On the other hand, the glue made the fabric sparkle a bit, which was an unexpected bonus.

If you are disappointed in the post so far, never fear, there’s more. Recently a video was going around about flexagons. It made me think of the old fortune tellers we made in grade school (I’ve also heard them called cootie catchers) and I decided to make one. I was ambitious and cut out pieces for a dodecahexaflexagon (twelve faces, each a hexagon, two of which show at any given time), but came to my senses and made a trihexaflexagon first. I cut six equilateral triangles of each of three decorative papers, two inches on a side, and a bunch of half inch squares of plain paper for the hinges. I glued them together using rubber cement, following the instructions on the Flexagon Portal, with only a front side on one end. After folding the whole thing up I glued the back on.

Changing from the second face to the third…

And back to the first…

Even more recently I discovered an old flexagon I made in middle school (I think). I’m not sure whether it officially counts as a flexagon, but it has more than two faces. In fact, it has four, and each consists of six squares in a two by three rectangle. With Thanksgiving I did not have time to explore it for this post, but it will appear at some point in the future!

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Since it turned into the kind of flower it did, I made a calyx for it. If you’re making a calyx you probably want to leave the loose ends of the flower yarn hanging out the back center of the flower. Each sepal is a chain with stitches down it, and this works best (stays flattest) if you stitch into only the top loop of the chain. Make a slip knot. *ch 7, and starting in second ch from hk, sl st, sl st, sc, hdc, hdc* five times (each time you’ll have a ch left over). Sl st to join and then sc around the inside opening, one sc per sepal (5 total). Put the loose ends of the flower yarn through the center of the calyx, stitch them through a loop and tie them together. Braid them with the initial end of the calyx and sc onto that braid with the working end of the calyx yarn. You’ll need to tighten it down on the braid and have the top of the stitches spiral around the braid to make it stable and straight.

[Alternatively, of course, you can make a stem however you like, or just finish off the yarn and have a brooch-style flower.]

The Amigurumi Army mission for July was nerdy crochet. I thought about something from a fandom, but couldn’t come up with anything I wanted to make. However, as we know, I am mathematically minded, so I looked in that world and found this:

A binary tree.

I made it from the top down, sewing as little as possible: when the second piece of each pair was made I just continued into the next segment down, stitching around the first piece without a gap. This required just a little thought about the order of operations. The only significant sewing was the leaves, though that was pretty significant. The smallest bits are 5sc in a magic ring, continued without increase. Then I just put pieces together and stitched around without counting, trying to keep things fairly compact, which is why nothing is exactly symmetric after that. The whole shebang is held up by eight pipe cleaners, one inside each of the smallest branches.

I finished it while visiting a friend with a jewelry tree, so I asked them to pose together.

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