Three quarters of a unit square placed at the origin are shaded as shown. The remaining quarter is treated similarly but mirrored, and so on to infinity. What are the coordinates of the vanishing point?

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## Solution

The vanishing point is at (2/3,2/3).

From reflection symmetry in the diagonal, it is clear that x=y. Now if you look at the top vertex of the three-quarter shape, its x-coordinate goes from 0 to 1 to 1/2 to 3/4 to 5/8 and so on as the shape gets smaller. This has the pattern 1-1/2+1/4-1/8+…. In other words it is the sum of (-1/2)^{n} with n=0,1,… This is called the Geometric series and it tends to 1/(1+1/2)=2/3 as n goes to infinity.

## Poem

The rabbit in it’s hole

With it’s ears so tall

And his eyes so small

The rabbit is jumping

With it’s twinkly nose

Here and there a jump he goes

Run run to that hole

Leading to Alice’s wonderland

And no one to understand

In deep soft sand

With tunnels dipped so deep

No end !