The ambidextrous moving sofa problem is to find the planar shape of maximal area that can negotiate right-angled turns both to the right and to the left in a hallway of width 1. The current best known solution was found by Dan Romik, and is shown here.
The small stellated dodecahedron, drawn here using Robert Webb’s Stella software, is made of 12 pentagrams, or 5-pointed stars, with 5 pentagrams meeting at each vertex.
This animation by Marshall Hampton shows the involutes of the curve \(y = x^3\). It lies at a fascinating mathematical crossroads, which we shall explore in a series of three posts.
The moving sofa problem asks: what is the shape of largest area that can be maneuvered through an L-shaped hallway of width 1? This animated image made by Claudio Rocchini shows one attempt to solve this problem.
A deltoid is a curve formed by rolling a circle inside a circle whose radius is 3 times larger. Similarly, an astroid is a curve formed by rolling a circle inside a circle whose radius is 4 times larger. The picture here, drawn by Greg Egan, shows a deltoid rolling inside an astroid. It fits in a perfectly snug way!
This image, drawn by Xah Lee, shows a deltoid and its catacaustic. The deltoid is the curve traced by a point on the perimeter of a circle that is rolling inside a fixed circle whose radius is three times as big. It’s called a deltoid because it looks a bit like the Greek letter delta: $\Delta$. The catacaustic of a curve in the plane is the envelope of rays emitted from some source and reflected off that curve.
This image, drawn by Greg Egan, shows a cardioid and its catacaustic. The cardioid is a heart-shaped curve traced by a point on the perimeter of a circle that is rolling around a fixed circle of the same radius. The catacaustic of a curve in the plane is the envelope of rays emitted from some…