From Finite to Infinite Volume, Part 2

Illustration for the KIAS Webzine HORIZON:
“Rigidity in 3D Hyperbolic Space: From Finite to Infinite Volume, Part 2”

This piece visualizes the concept of boundary maps using two monkeys as characters: Kim the monkey and Dong the monkey.

On the left, Kim the monkey independently finds fruit, acting as the agent of choice. On the right, Dong the monkey looks toward Kim and follows his directions to find fruit, showing a lack of autonomy. The fruits they encounter along the way are illustrated like collectible game items.

As Kim (left) makes decisions, Dong (right) follows them, passing through a path of grape → apple → banana → cherry. The arrows marking Dong’s path are grayed out to represent his passive role in following Kim’s choices. One specific arrow is highlighted in red to emphasize the directive relationship—Kim points, and Dong follows.

Excerpt from the article:

"There are two trees. These are exceptionally kind trees that bear all kinds of fruit—apples, pears, watermelons, oranges, and more. Two hungry monkeys attempt to climb a tree each. Let’s call them Kim the monkey and Dong the monkey.
As they climb, there are moments when they must choose which branch to take. Dong, being a very obedient monkey, always follows Kim’s directions at each decision point (although Kim’s own choice may differ from what he tells Dong to do).
Eventually, both monkeys will reach some kind of fruit. For example, Kim might arrive at an apple while Dong ends up at an orange; or Kim might reach a pear while Dong finds a watermelon.
Now, are the fruits that Kim and Dong reach independent of each other? Not quite—because Dong moved according to Kim’s guidance. Thus, the fruit Dong ends up with can be considered a kind of ‘function’ or ‘map’ of Kim’s final outcome."