Ever wondered how a doughnut's shape is mathematically understood?
Ever wondered how a doughnut's shape is mathematically understood?
Imagine you're designing a new kind of ring-shaped coaster that perfectly fits a circular table. You want to ensure it wraps smoothly around without any gaps or overlaps.
Picture the table's edge as a circle. The coaster's shape is like a doughnut, with a hole in the middle. This doughnut shape is what mathematicians call a torus. The torus helps us understand how to create a smooth, continuous edge for the coaster.
Example
If the table's edge is a circle with a radius of 5 cm, the coaster's doughnut shape will have a major radius (distance from the center of the hole to the center of the doughnut) of 5 cm and a minor radius (radius of the doughnut itself) of 1 cm.
Remember this
The torus concept helps designers create smooth, continuous shapes for objects like coasters.
Text adapted from Wikipedia, licensed under CC BY-SA 4.0.
Riemannian manifold
Riemannian manifolds generalize Euclidean space concepts like distance and curvature
Manifold
A manifold locally resembles R^n
the Lp norm ball shape changes as p goes from 1 to 2 to infinity
How does the shape of a ball change as we measure distance differently?
Curvature
Curvature measures the angular rate of change of the direction of the tangent line per unit distance along the curve
Open set
How can we understand closeness without measuring distance?
the L1 unit ball is a diamond shape and the L2 unit ball is a circle
Why does a ball look different in various dimensions?
Swipe through 100 ML concepts daily
Open Pocket Polymath