In this episode several different mathematical topics are mentioned including network valuation theory and area filling problems, but we're
going to focus on Euclid's Orchard.
Euclid's Orchard is a thought experiment involving an imaginary forest of trees. Each tree is an idealized line segment, having length 1, no width or depth, and standing straight up. This forest is on the two-dimensional coordinate plane, and there is a tree standing up at every integer lattice point on this coordinate plane.
Activity 1: Looking into Euclid's Orchard
Imagine that you cut away the tree at the origin and stand at the origin at a height of zero, looking at the trees in the first quadrant (the trees whose x and y coordinates are both non-negative). If one tree is on the line of sight between you and another tree, then the tree in the middle obviously blocks your view of the farther tree.
Which trees can you see from your vantage point? All trees are at coordinates (n,m) for some integers n and m. If you can see a tree at location (n,m), what can you say about n and m? If you can't see a tree at location (n,m), what can you say about n and m?
How far away is the base of the tree located at (n,m) from you?
What is the angle of extent of the tree located at (n,m), supposing that you can see it?
If two trees are standing at lattice points (n,m) and (p,q), how can you tell if the tree at (n,m) is seen to the right, to the left, or on the same line of sight as the tree at (p,q)? What are the conditions on the integers n, m,p, and q?
Using the answers from above, draw a picture of what you would see looking into the first quadrant of Euclid's Orchard from your vantage point at the origin. Would the other quadrants look any different?
Dirichlet's function, which we will denote by f, is a function from the interval to the real numbers. Dirichlet's function is defined by the following:
f(x) = 0 if x is irrational
f(x) = 1/s if x is rational and in lowest terms x = r/s
Activity 2: Graphing
Graph Dirichlet's Function.
On your graph of Dirichlet's function, draw a vertical line segment connecting every point of the graph not on the x-axis to the x-axis.
Compare this modified graph of Dirichlet's Function to the sketch you made earlier of the view inside Euclid's Orchard. Are they similar? Why? Are they the same? If they are not identical, is there a transformation you can perform to make them identical?