Here are some shadblow (Amelanchier canadensis) or serviceberry leaves that I collected from my yard this past Fall. Their tattered appearance speaks of the heavy use that the local ecology has made of them. Not only did they provide sugars for the mother shadbush, but insects and perhaps other arthropods (such as mites or spiders) have used them for food and habitat.

1. Each leaf is spotted with many skeletonized areas. There appear to be two sets of skeletonizations. One set has light-colored edges, the other has dark-colored edges. Anyone have an explanation?

2. There’s a tiny web of silk on one of the leaves. Who made it?

3. Stuck in the silk are some small dark objects. Are these frass (insect detritus) from the maker of the silk?

]]>I have two learning “disabilities” (“learning differences” is a more politically correct, and more accurate term): dyslexia, and ADHD. Some people have been surprised to learn this. I am among them. I was only diagnosed with these learning differences recently.

I think of dyslexia as a different brain architecture than the norm. Dyslexic brains are wired in such a way that distant parts of the brain are well connected at the expense of dense connections locally. This different brain architecture makes for a different profile of strengths and weaknesses from more common brain types.

Many tasks that are accomplished by non-dyslexic brains automatically and unconsciously, are only accomplished with conscious effort by the dyslexic brain. Examples of such tasks may be (and this varies individual-to-individual) reading, understanding speech, and performing repetitive physical tasks.

On the other hand, dyslexic brains tend to be good with spatial reasoning, seeing connections, recognizing ambiguity, and thinking in narratives.

For most people, dyslexia does not mean a complete inability to read. It does usually means a slower process learning to read, and some lifelong level of difficult reading as compared with the norm.

My list of symptoms is quite long. I’ve decided to write an entirely different post about it on my non-math blog.

As a dyslexic, I’ve never been good at calculations or recalling rote facts like times tables. Here’s the thing: beyond a certain point in mathematics, it’s not really about calculations. Sure, calculations can help big time in certain mathematical disciplines–Gauss was known for being an amazing lightening calculator–but there are areas of math where it’s possible to stay comfortably far from computation.

As I said before, dyslexia comes not only with weaknesses, but with strengths*. Some of these strengths play well into mathematical thinking, such as:

- Spatial reasoning. Dyslexics tend to be good at thinking relationally in three dimensions. This is great for many areas of math. Topology is one of those. My PhD is in topology.
- Seeing connections. Progress in mathematical research is made by drawing connections between disparate subfields. Dyslexics often have a strength here also.
- Thinking in narratives. The way to predict how well a child will do in math is by how complicated a story they can tell. See this ScienceNews article. It makes sense: proofs are the bread-and-butter of math. Proofs are really a lot like stories. Dyslexics are usually good narrative thinkers.

I really didn’t like math at all up until I hit geometry in ninth grade. That’s when math became much easier for me. It may be confusing that moving into more advanced mathematics actually made math easier for me. The key to remember is that my brain works differently from the brains that school curricula were designed for.

Here’s a concrete example of how my brain works differently: concept precedes process. A friend was over for dinner the other night. She said that the part of geometry that made sense to her was finding the areas of regions by breaking them into simpler shapes. She said she could do this because of her experience sewing. See what’s happening there? Sewing is a process. Through doing the process my friend was able to get the concept of computing areas. This is a pretty normal way of learning: do a process by rote procedure, then understand the concept. My brain doesn’t like that kind of learning. I learn in the opposite direction: concept first, then procedure. This is also how my dyslexic daughter, Lucy, learns. Lucy had a hard time learning to count. Because my brain works so similarly to hers, I was able to get at the heart of her counting difficulties. The punchline is that she understood the concept, but had a hard time with the process. I wrote about it in a previous post. Mathematics at the PhD level is not about learning process by rote. In the abstract world of pure mathematics, concept may be all you have.

Like I said, geometry class was when math became interesting, and easier for me. Suddenly I was in a world, not of strands of symbols to be processed, but of shape, space, lines, angles, concepts, and narrative-like proofs. Suddenly everything made sense.

Most exciting of all in my high school freshman geometry class was the idea of non-Euclidean geometry. We learned about the geometry of the sphere (a sphere is the surface of a ball, in math-speak). I raised my hand in class and said “if two-dimensional spherical geometry is like the surface of a three-dimensional ball, maybe our universe is like the surface of a four-dimensional ball.” The teacher seemed to think that I was getting a bit ahead of myself. Maybe I was, but I was finally swimming in my element.

Other geometry students were doing things like memorizing theorems. Memorization made no sense to me. Do you know which way is up by memorization? Do you know that snow is cold by memorization? Geometry, and later calculus, topology, and other math classes just made intuitive sense to me. It’s not that I didn’t have to try. But it felt like what my brain was meant to do. In graduate school for instance, I could sit by the pool while my kids played and let my math problems roll around in my head. My brain naturally attacked them with it’s native strengths of spatial reasoning, finding obscure connections, thinking in narratives, and taking different points of view. Later that evening when my family was asleep, I would have to tease my thoughts out into sentences, mathematical expressions, and other strings of symbols.

Unfortunately for me, if you go far enough down the path of any geometrical problem, the tools of algebra start winning out. According to a quote attributed to Michael Atiyah:

Algebra is the offer made by the devil to the mathematician…All you need to do, is give me your soul: give up geometry

That’s the reality that I hit at the end of my graduate career. It’s the reality that I live with as a career mathematician. I’m not yet willing to sell my soul. My brain still wants to swim in its native stuff.

*I’m stealing this list of strengths from The Dyslexic Advantage by Eide and Eide. ]]>

After a brief walk around the back-yard woodlot yesterday, here’s what ended up in my pocket.

]]>Here’s something truly amazing that sprouted recently in my backyard. They are plants called Indian pipe, or ghost plant, or corpse plant. The Latin name is *Monotropa uniflora.* It has no green to it because it has no chlorophyll! No chlorophyll means no photosynthesis. This plant is a parasite. It derives its energy from a fungus which in turn get’s its energy from a tree (probably the American beech in my backyard). Here’s a great little article about M. uniflora.

Think about it: an elegant, but ghostly white little flower living out it’s life cycle like so many other flowering plants. It’s pollinated by bees, produces seeds, germinates and grows in the rich forest soil, but it makes it’s living so very differently from other plants.

Here’s a scholarly article whose abstract is worth a quick read. It challenges the idea that parasites are “all bad.” A quick quotation: “Parasites and pathogens act as ecosystem engineers, alter energy budgets and nutrient cycling, and influence biodiversity.” And another article:”parasites may be the thread that holds the structure of ecological communities together.”

And if you’re looking for your math fix: quantitative parasitology is a thing, yo.

]]>Our next parabola video. All parabolas are the same shape, just different sizes!

]]>In this second installment we multiply with parabolas!

]]>My kids and I are working on a series of videos about parabolas. Here’s the first installment.

]]>More wintry weather for us tonight and tomorrow. While we wait for new spring leaves, here are some more leaves from my yard last fall.

1. Tar spot looks like tar on the leaf, but is caused by a fungus, Rhytisma acerinum. This is an endophytic fungus: it lives inside of the cells of the leaves that it feeds on, without really causing much damage. See this paper

2. Don’t maples have a nice star shape? Japanese maples are particularly nice, but of our native American maples, I really like the shape of these silver maple. Check out this highly mathy article on leaf shapes.

3. I’m not sure what has caused these spots on the stems. (Remember from an earlier post that these stems are called petioles.) At the spot on the right there appears to be an incision in the leaf, perhaps made by an insect for deposition of it’s eggs?

]]>My ten-year-old daughter asked me for help doing a mathematical STEM-fair project. We had a lot of funny doing the project together. Here is a brief description of what we did.

We decided on the following question: “How many decominoes can be filled in (tiled) with a pair of pentominoes in more than one way?”

There are 12 pentominoes. I had Abby figure out the number of *pairs* of pentominoes. She reasoned that since you have 12 choices for the first pentomino in a pair, and 11 choices for the second (but switching first and second you’ve counted each pair twice) there are 12×11/2 = 66 pairs. There are 4,655 decominoes. So we had 66 x 4,655 = 307,230 trials (trying each pair against each decomino). Abby wasn’t going to be able to do this by hand. Instead I taught her the Python she would need to program our computer to do it.

We couldn’t find an explicit list of all 4,655 decominoes anywhere, so the first order of business was to write a program to generate those. Actually, before that we had two prerequisites to understand. **First** I had to introduce her to the very basics of programming: what is a variable and how do they work? What are loops? How do they work? How about lists? Functions? There was some frustration on both our parts, but we got it hammered out. **Second** we had to decide how we would represent polyominoes in the computer and how we could manipulate them. I had already done some Python projects with polyominoes, and was representing polyominoes as lists of cartesian coordinates of each square. Abby understood the representation immediately, saying “oh! It’s like a coordinate grid!” I had Abby figure out (and write functions for) how to flip and rotate polyominoes.

Abby largely wrote the code to generate the decominoes herself*. Here is a snippet:

def poly_change(n_ominoes): n_plus1_ominoes = [] for p in n_ominoes: for square in find_adjacent(p): new_t = p + [square] new_t = slide_to_zero(new_t) new_t.sort() if is_not_in(new_t, n_plus1_ominoes): n_plus1_ominoes.append(new_t) for p in n_plus1_ominoes: fancy_print_board(p) print p print len(n_plus1_ominoes) return n_plus1_ominoes

Here’s how she bootstrapped from tetrominoes to decominoes. I suggested using a for-loop, but she preferred this way:

S1 = [(0, 0), (0, 1), (0, 2)] B1 = [(0,0), (0,1), (1,0)] trominoes = [B1,S1] tetrominoes = poly_change(trominoes) pentominoes = poly_change(tetrominoes) hexominoes = poly_change(pentominoes) septominoes = poly_change(hexominoes) octominoes = poly_change(septominoes) nonominoes = poly_change(octominoes) decominoes = poly_change(nonominoes)

At each step from n-ominoes to (n+1)-ominoes she ran the code and checked that the number of polyominoes found by her program was the same as what we could find reported online.

We stole code for finding all tilings of a given shape (our decominoes) with a given set of tiles (our pairs of pentominoes). I had also written some wrapper code for it previously. Here’s (a bit of a simplification of) the main loop Abby wrote to do each of her trials:

for d in decominoes: for pop in combinations(Pentominoes, 2): tile_pair(d, pop)

Of the 307,230 trials only 3,486 (about 1%) had any tilings at all. Of those 3,486 trials that could be tiled, only 41 (again, about 1%) could be tiled in more than one way. Each of those 41 could be tiled in exactly two ways (none could be tiled in three or more ways). Of the 41, 39 had a reflection symmetry (and this symmetry was what caused there to be two tilings). Here’s an example of one of those 39:

The other two trials that could be tiled in two ways, but have no symmetry are the ones that I think are most interesting. These two are shown at the top of the post. Here they are again:

I call them Abby’s puzzles, thinking of them as a board to fill in with a pair of pieces. We had tons of fun! I hope that Abby continues learning math and programming.

*I had written find_adjacent() and fancy_print_board() for a previous project. is_not_in() took some thinking, but again, it was largely Abby who did it.

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