Tuesday, January 26, 2016

the triangle of triangles

Is there a triangle that contains every possible triangle? In one sense, yes there is.

In "The Stiener-Lehmus Theorem" (which appears in The Best Writing on Mathematics, 2015, edited by Mircea Pitici), John Conway and Alex Ryba introduce a clever nomogram which they call "the triangle of triangles."

As the authors describe it,
each point corresponds to a triple of numbers A, B, and C, that add to 180, and so to a shape of a triangle. In the figure, A is constant on downward sloping lines, B on upward sloping ones, and C on horizontals.
In the diagram below, the point P provides us with a triple (36, 60, 84), which corresponds to the shape of a triangle ABC with those angles. Note that A and B are easier to read off the figure than C, so it may help to add an additional axis.

To say P corresponds to the "shape" of a triangle, it is meant that P represents an infinite family of similar triangles whose angles (measured in degrees) equal the numbers given by the triple described by P.

Let's see where some favorite triangle families live on the triangle of triangles. In the simplified figure below, green lines are drawn at 90 and light blue lines are drawn at 60. These intersect at the special 45-45-90 right triangles.

And here are some more favorites. With the light green lines drawn at 90, the dark green drawn at 60, and the orange lines drawn at 30, we get the 30-60-90 special triangles, and equilateral.

Maybe it is a little surprising that the triple you get from the points within the triangle should sum to 180. In the diagram below, those values correspond to the lengths of RC, QC and PS, with appropriate scaling factors applied

If for the sake of simplicity, we assume that the lengths of AP and BC are 1, things simplify quite a bit. The relationship to be proven reduces to what is shown in the figure below.

One way to show this is to start with the median DC, and show how the required lengths are related to it.

Of course, it makes sense if you are drawing a triangle-of-triangles, you'd likely want to draw it as an equilateral, but as you might notice, any old triangle seems to do (with the right scaling). The figure below shows one set of measurements taken in GSP, where you can distort the triangle and convince yourself that the relationship holds generally. Feel free to re-do the proof above for the general case :)

Thursday, December 17, 2015

a universe of puzzles

In The Puzzle Universe: A History of Mathematics in 315 Puzzles (TPU), Ivan Moscovich stretches the concept of puzzles to encompass almost anything that combines curiosity and playfulness (playthinks is his preferred term for this more general category of puzzling items). No surprise - these playful curiosities are inherently mathematical. In an informal and accessible way, Moscovich details the development of these puzzles, revealing their surprising family resemblances and the deep mathematics behind their playful exterior.

An example of a playthink that might not at first inspection resemble a puzzle in the usual sense is the spirolateral (TPU puzzle 270). Drawn following basic rules, a family of interesting geometric objects emerges. The rule for drawing the third spirolateral is to draw a line 1 unit long, turn 90 degrees, draw a line 2 units long, turn 90 degrees, draw a line 3 units long, turn 90 degrees, and repeat. The general rule is to draw lines starting at 1 unit long and going up to n units long, turning 90 degrees between each line, and then repeating the process from 1 again. A natural thing to try in LOGO, the first few are shown below.

the first few spirolaterals

The puzzle-prompt is "what do the next two look like?" A natural question follows... what do they all look like? In answering the first question, you are following rules, in answering the second, you are using mathematical thinking.

a later spirolateral

TPU also demonstrates how to to read almost any mathematical object as a puzzle, which opens up pathways of inquiry of surprising richness. In Moscovich's treatment, a decorative pattern (see here and here) can become a puzzle (as in TPU puzzle 83) simply by asking questions like, "how many separate loops make up the image below?"

a decorative pattern

One of the pleasures of "recreational mathematics" is all the little discoveries you make about the simple patterns and relationships you are exploring. Reading TPU, you may find out that your recent breakthrough has been well known for a long time, was the basis for a 19th century children's game, or was proved by Gauss when he was ten. Finding out that the puzzles you've been playing with (and your inventive solutions) have a long history is a nice reminder that while you may be working on puzzles alone, you're not alone in working on puzzles. A while back, I was playing with punctured chessboards and knight tours - from TPU puzzle 100 I learned that an Italian mathematician devised a knight puzzle on a 3x3 punctured board in 1512.

a knight walks around a small punctured board

There are many familiar items in TPU: Pascal's triangle (TPU 119), the river crossing problem (TPU 73), liars and truthers (TPU 306, 307, 308), the Monty Hall problem (TPU 293),... I could go on. Something familiar, and something new can be found every few pages. Even for many of the more familiar items, Moscovich often has an unexpected connection to present. For example, I knew that the midpoints of any quadrilateral would form a parallelogram (TPU 130), but had not known that, in general, derived polygons formed by joining midpoints tend to become "more regular."

a derived parallelogram, some derived pentagons

While diving into the alternately obscure, well-known, and in some cases apocryphal origins of many puzzles and the mathematics associated with them, TPU provides a welcome counter-narrative to utilitarian accounts of the development of mathematics. More comprehensive historical treatments reveal, however, that the history of mathematics is richer (and stranger) than even TPU suggests. In some ways, a better title would just have been simply "mathematics in 315 puzzles," as TPU provides a gentle and accessible introduction to the essence of mathematical thinking.

The Puzzle Universe is a quixotic, informative, and enlightening encyclopedia of recreational mathematics. It should prove to be an inspiration to mathematical idlers, and a rich resource for learners and teachers who wish to be attuned to the playful and creative side of mathematics.

Thursday, November 19, 2015

Thursday, October 22, 2015

more pentagons, decagons, stars and rhombs

Kept adding on to one of the pentagon patterns from the previous post...

Eventually, ended up with  rough decagon shape - how many little decagons in there?

Zooming in, there are some nice patterns in there:


Thursday, October 15, 2015

out from a ring of pentagons

Looking at the Kepler pentagonal tiling, it seems there is skull staring back, warning me against spending too much time playing around with polygons.

But regular pentagons encourage time wasting by how they fail to fit together, forcing fusings and overlaps. For example, finding a way to build out from a central ring of ten pentagons, you might try a ring of ten fused decagons.

After another jagged ring of 30 pentagons, you can add another ring made of ten rings of pentagons joined by hexagons (formed from fused pentagons).
One more step, and the decagons seem to be drifitng outward - once overlapping, now with stars between them.

Or, you could start with a central ring of pentagons surrounded by five stars and five rhombs.

A slightly more compact ring of pentagon rings comes next.
And as they pull apart - more stars.

Just remember, those aren't crania, just decagons and pentagons.