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New geometry discovery could stop lunar landers from falling over

Meet Bille, the name given to the world's first monostable tetrahedron—a four-faced object that will always land on the same side, no matter its starting position. This feat of geometry and engineering solves a nearly 60-year-old mathematical mystery and could help in designing self-righting spacecraft for future lunar or planetary missions.

In 1966, the eminent British mathematician John Horton Conway and his partner, Richard Guy, wondered whether it was possible to construct a tetrahedron made of uniform material with an even weight distribution that would always flip to its stable side.

They believed that an unevenly balanced monostable tetrahedron was possible, although they could never prove it.

So the mystery remained unsolved until Professor Gábor Domokos at the Budapest University of Technology and Economics (BME) and architectural student Gergő Almádi began working on the problem three years ago.

Mystery solved

Using powerful computer models, they developed a theoretical framework. They realized that a monostable tetrahedron capable of always landing on its stable face on a flat surface would have to be mostly hollow. And one side would need to be thousands of times denser than the others.

Working with a Hungarian precision engineering company, they created the world's first physical model of a monostable tetrahedron—a skeleton of lightweight carbon fiber tubes with one side made from a high-density tungsten-carbide alloy.

The structure measures 50 centimeters on its longest side and weighs 120 grams. It was unveiled at BME, while details of the discovery were recently on the arXiv preprint server. The model was nicknamed Bille, after the Hungarian word billen, meaning "to tip." No matter which face (A, B, C or D) you start with, it will always settle on face D.

Improving lunar lander design

One possible application of this research is improving the design of lunar landers so they can right themselves after falling over. This recurring problem has brought several missions to a premature end, such as the IM-2 lunar mission earlier this year, when the uncrewed Athena spacecraft fell on its side in a crater.

Domokus and Almádi are hopeful their work can help. "While it may not be possible to design objects which can passively self-right on any terrain, designing for self-righting on a horizontal support may be feasible and we hope that for those designs our study could offer insights."

Beyond space, the research could also inform the design of other self-righting objects, such as legged robots navigating challenging terrain.