# The typographic scale

The typographic scale has been used for centuries to choose harmonious font sizes. It has been likened to a harmonious musical scale. But there are flaws in those historical values.

# The egg problem

The egg problem is a classic programming challenge. However, there is a mathematical solution. You can solve this problem analytically, even in the general case with $$n$$ eggs.

# Approximate a circle with cubic Bézier curves

Bézier curves are often used to generate smooth curves because Bézier curves are computationally inexpensive and produce high-quality results. The circle can be closely approximated using a cubic Bézier curve that is constructed to minimize the radial deviation from a circle.

# Serialize a binary tree in minimal space

The natural way to serialize a binary tree requires $$2n$$ bits to store a binary tree with $$n$$ nodes. However, it’s possible to reduce this to $$2n - \frac{3}{2} \log_2 n - \frac{1}{2} \log_2 \pi$$ bits, with the help of the Catalan numbers. This is the minimal space needed to serialize an arbitrary binary tree with $$n$$ nodes.

# Symmetric senary

Symmetric senary is a tiny numeral system which is intended for people to use in their everyday life. It is especially useful for dividing things up in practical ways, and easily estimating things like tips and mileage without ever needing a calculator.