I’m Andrew Hogan. These articles begin with ordinary things: a bouncing ball, a drive, a triangle, the steps of long division, or two people sharing a moment. Each offers a question to try and some thoughts about where it might lead.
You can work through an exercise, follow a construction, or simply read. Bring your experience. There may be something here that looks different from where you are standing.
Getting to know you: What if Everyone met everybody?
Two people share three minutes. Imagine doing that until every living human being had shared exactly one moment with every other.
That process would complete in 47,491 years. It’s the minimum in one idealized version of the problem, at least one that just rocks. The arithmetic has a solution. On the table is a time value for maximally momentous opportunity. The human question is less easily settled.
Read The Humanity Moment Clock →
Thinking about thinking
A population, a shared moment, and a clock in the sky. What happens when the same little recipe is tried in another time? This exercise follows the question through history and celestial cycles.
There are calculations to check, connections to explore, and an invitation to continue the investigation on your own. Thinking may be dangerous work. Adult supervision may be required.
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How long can a ball keep bouncing?
A ball bounces on a piece of stretched balloon rubber covering a glass. It rises, falls, and rises again.
Watch the height of each bounce. What changes? Where might the energy go? What would have to be different for the bouncing to continue forever?
A small experiment can give us something concrete to observe and something much larger to imagine.
A question with everyday consequences
How much room does it take to stop?
At highway speed, a car travels a considerable distance while its driver notices a hazard and responds. Braking takes more distance still.
But stopping before a stationary obstacle and following another moving car are different situations. What assumptions connect them—and where do those assumptions stop helping?
Explore the articles
Time and the numbers we live with
The Humanity Moment Clock
How long would it take for everyone to share a moment with everyone else?
Your One Billionth Birth-Second
What becomes noticeable when we count our lives in seconds?
Where Do You Draw the Line?
A drive, a traffic report, and a few minutes saved.
Moments: Winding Back
Shared time, old questions, and a game of thinking about thinking.
Making something from a possibility
The Forbidden Triangle
Two sides, a missing length, and a tent made from unequal halves.
Robert’s Remainder
Long division leaves a trail of intermediate results.
Motion, limits, and physical models
Car Lengths
Reaction, braking, and the limits of a simplified model.
Tight Turn Limit
A bicycle, a turn, and the grip between tires and road.
Experiments to notice
Balloon Over Glass
A bouncing ball opens a question about energy and forever.
Charged Sphere Pump
What can repeated redistributions of charge accomplish?
A way to explore
Most articles follow a familiar path: an exercise, an explanation, and some thoughts or further questions.
Before looking at an answer, you might make a prediction. While reading, notice which conditions are given and which are assumed. Afterward, try changing one condition. Does the conclusion still follow?
Some questions are easier to investigate with another person. A game, a measurement, or a small experiment can become a shared moment. The mathematics may help arrange such opportunities; people give them meaning.
Sometimes a useful result is an answer. Sometimes it is a clearer question, a corrected calculation, or a new thing to make.
Keep the conversation going
These pages grew from teaching, conversations, personal experiences, and the contributions of students. I try to give those contributions room to travel.
If you find an error, a missing assumption, a different construction, or a question worth adding, I would like to hear about it. Pointing to a particular sentence or step helps me understand what you noticed.