Plants without photosynthesis

IMG_1171Here’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.
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Yay space!

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Parabolas III

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

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Parabolas II

In this second installment we multiply with parabolas!

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Parabolas I

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

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Silver Maple Leaf Ecology

Silver Maple Leaf Ecology

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?

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Abby’s Puzzles


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)
            if is_not_in(new_t, n_plus1_ominoes):

    for p in n_plus1_ominoes:
        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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