Rectangle PQRS has X and Y on the edges. Triangles PQY, YRX and XSP
have equal areas. Prove X and Y divide the sides of PQRS in the
Straight lines are drawn from each corner of a square to the mid
points of the opposite sides. Express the area of the octagon that
is formed at the centre as a fraction of the area of the square.
Find the ratio of the outer shaded area to the inner area for a six
pointed star and an eight pointed star.
It seems very possible that music (and mathematics) has its own
way of talking and it's easy to feel on the outside of that
conversation. The encouraging thing is that mathematics can help us
understand music, and maybe the other way round too.
This is the first of three problems based around this idea (the
other two are
Pythagoras' Comma and
Equal Temperament ):
The Greeks, Pythagoreans especially, were interested in the
notes made by plucking a collection of strings of different lengths
(I'm guessing that the strings all had the same tension - maybe by
hanging equal weights, beyond the section being tested).
Now making music on two strings is a bit limited, so what they
tried to find was a collection of lengths that would all sound good
together. For convenience, no length would be more than double the
shortest length. They settled on a six point scale (a set of
agreeable notes). We'll call the longest length Note 1, and make
that length our unit. The shortest length (half a unit) we'll call
The length which makes a ratio of two to three with the length
for Note 1 turned out to be the fourth note in their scale.
What might be good fractions for the Notes 2, 3, and 5 ?