1. Pursuing, extending, and refining an idea, just because one can.
Why Math?
Pursuing, extending, and refining an idea, just because one can.
Sometimes an idea pops into our head and we’re just curious to see how it plays out.

2. F(n) = n2 If n = 2, F(n) = 4 If n = 3, F(n) = 9 If n = 4, F(n) = 16 …
Here’s an elementary idea -- I’ve pictured it as a machine. We toss a number in, and some rule tells what number comes out. This machine multiplies a number by itself: if we put 2 in, 2 times 2 comes out; if 3 goes in 3 x 3 comes out, and so on. We can, of course, put in any number we like and in any order we like. So let’s make an additional restriction and see what happens.

3. Iterating F(n) = n2 If n = 2, F(n) = 4 If n = 4, F(n) = 16…
What if we say that whatever comes out must be the next number that goes in? For example:

4. Too prosaic! Math is not just about numbers
What if we put a line into the machine and got out a bent line?

5. V(—) = °
Well, of course, to make a clear rule, we have to say something about *how much* bend we put in. Here the 22° refers to how much we turn to the left before we start the first leg of the V.

6. V(—) = °
22° between the horizontal and the first side of this upside down V.

7. V( ) =
If we put the *V* back in, each of its legs gets bent into a V, following the same rule.

8. V( ) =
…and this is the result. But remember that the long, seemingly-one-piece top, is really two separate pieces. (Flip back to previous slide.)

9. V( ) =
Again, replace those straight parts with Vs.

10. V3(—) °

11. V4(—) °
And again.

12. V5(—) °
And again.

13. V6(—) °

14. V10(—) ° A cauliflower?

15. V10(—) ° Or broccoli?
By increasing the amount of bending at each step -- by making a pointier “V” with 26° instead of the 22° we were using before -- we get a similar structure, but, well, bendier. Is it pure coincidence that the rule for the cauliflower and for the broccoli are essentially the same rule? Or is it an important clue to some biochemical (or other) reality that we could look for scientifically? That is, in the mathematical game we’ve played (quite by accident!), we see that changing a *single number* in an otherwise *identical rule* allows us to generate shapes that visually match these two closely related plants. Is that a clue that something just as “simple” exists in the physics or biology of these plants, something that we might not otherwise have thought to look for?

16. V10(—) °

17. V10(—) °
When we increase the angle even more, yet other shapes appear. If we increase them enough, they begin to feel a bit less “biological” (though this example still has a bit of the sense of biology left).

18. V10(—) °
It does sort of make sense, now that we think of it, that extreme angles would be a surprise in biological objects.

19. V10(—) °
And we can reduce the amount of bending again.

20. V10(—) °

21. V10(—) °

22. V10(—) °

23. V10(—) ° A muffin?
Or the brain? Or a brain coral? Except for the extreme regularity (but we could eliminate this with a bit of randomness tossed in), this also looks a bit like a portion of the surface of certain kinds of clouds.
The real question is: why does this look like *anything* we’ve seen before?! It is, after all, just the result of an arbitrary mathematical rule applied mechanically. Does the fact that this looks like stuff in nature mean, perhaps, that nature, too, is applying rules “mechanically.”

24. V10(—) °
Reducing the amount of bending even more

25. Branching
B( ) =
B( ) =
The function “B” takes a line to a branched line.
If it is applied to a branched line, it treats each branch just as it treated the original stem.

26. A bent line
We could imagine starting with a slightly bent line, too, and applying a branching rule to it.

27. B ( )
Here is a slightly fancier branching rule, that adds five bent lines to the original bent line (the trunk).

28. B2 ( )
Each “branch” on that previous picture is a bent line, so applying the rule to each branch creates a little tree in its place. Look back a slide to see how this branched limb is just a tilted copy of the previous entire tree, scaled down to fit the bare limb that it replaces.

29. B2 ( )

30. B3 ( )
Applying the rule again, makes each twig in the previous picture into a little tree.

31. B4 ( )
It’s Springtime!

32. B5 ( )
And summer!

33. Roots?
Why does the root system behave (visually at least) just like the top branches?

34. Bronchi?
What do roots and bronchi have in common?

35. Why these shapes? Trees? Bronchi? Blood vessels? Roots?
Even broccoli and the tops of muffins?
Why not cats, pigs, and elephants?
Things that need to minimize absorption or loss (of water, chemicals) across their boundaries need to minimize the amount of surface for their volume. If you move around to get nutrients, you’d like to minimize the loss (e.g., water) while doing that. Of all shapes, the sphere has the least surface-area for a given volume. Animals are sphere-like to conserve water. So are cacti! Things that need to *maximize* absorption (of, e.g., sun, water, air) across their boundaries need to *maximize* surface-to-volume ratio. If you stay still and depend on the environment to reach you (plants, corals, alveoli in our lungs, capillaries), you build a branching structure. (other examples)

36. And why are these so convincing?
Does the fact that they look right actually tell us anything about reality? Or is this just “coincidence”? Remember… …people see bears in the stars!
The fact that we see pattern -- like the animals and people we see in the constellations -- doesn’t mean that nature “saw” the pattern!

37. Mathematical ways of thinking…
But when we consistently see the same patterns in different places, it *is* smart to ask ourselves what, if anything, *might* be the common cause or underlying structures.

38. An insight from mathematics
A very intricate structure can arise from an extremely simple rule.
DNA for “complicated” shapes?
DNA needs to pack a lot of structural information into a very tiny place. It would make sense not to bother with having an instruction for the placement of each limb, branch, twig, and leaf, and separate instructions for each leaf, and so on. Trees are, of course, *not* as “orderly” as these pictures, but the randomness does not need to be built into the *rule*. The rule may say at what intervals and at what angles and at what size-reduction to create branches, and weather, or bugs, or… may influence how these rules play out. Perhaps a branch budded, but a bug ate the bud.

39. Curiosity, ideas, and imagination
Mathematics can help us answer questions…
…but it can also help us ask them.