9.3 Gnomons

9.3 Gnomons or ... Nature knows!!!

Topics
Introduction
Self-gnomonicity
Square gnomons

Introduction

as a baby grows, does it grow new legs? another head?
if a small tree looks like this:

will it grow to look like this?

Background:

gnomons were discovered and studied by the Greek philosopher Aristotle
who noticed that older (and larger) living things
have approximately the same shape as their younger (and smaller) versions
etymological note: gnomon is Greek for "one who knows" (as in "gnome", also "carpenter’s rule")
a gnomon is a geometrical figure which

when attached to another geometrical figure A
results in a new figure similar to (having the same shape as) A

do you recall the mathematical definition of similar figures ?
Aristotle's thought: living things grow by pieces which are gnomons of the original

Example (from geometry, not nature):

A is the original rectangle; the ratio of its short side to its long side is s/l
its gnomon is L-shaped (like a carpenter's rule ???)
when you "add" the gnomon to the original rectangle, you get a larger, but similar rectangle
for which ratio of short to long is S/L
because the ratios of corresponding sides of similar rectangles are equal (by definition of similar)
the equation expresses (mathematically) the fact that A is similar to A-with-the-gnomon-added

Self-gnomonicity

Can a rectangle be its own gnomon? Consider:

original rectangle its gnomon??

Stack 'em up! Turn it on its side

what must be true for
this one . . .

to be similar to
this one?

Let’s follow through algebraically:

note that for the second rectangle:

short side = l
long side = 2s

and we want the ratios (short side)¸(long side) for each of the rectangles to be equal
here goes:

2s² = l²

so the long side must be (approx. 1.414) times the short side

Exercise: create a self-gnomonic rectangle with a short side =

the long side must be times the short side = = 2
so it's a rectangle with dimensions s = , l = 2
can you check such a rectangle for self-gnomonicity?

Square gnomons

When does a rectangle have a square gnomon? Here's a rectangle:

the rectangle

We can't attach a square gnomon to the short side (obviously). Here's one to attach to the long side:

its gnomon

Attaching, we have:

Remember, the ratios (short side)¸(long side) of the green and combined rectangles must be equal:

so we need to solve the equation
but have you seen such an equation before (where a and b had different names?)?
what kind of rectangle gave rise to such an equation?

··

so, it is ··that have square gnomons!!!