The mathematician discovered that the new "tumbler" shape will be designed as a toy

Similar to the tumbler
Similar to the tumbler

Indian Star Turtle
Indian Star Turtle

Two researchers at the University of Technology and Economics in Budapest, Hungary, found the same shape as the tumbler, but with no uniform weight inside. The scientific paper was published in the world famous magazine Mathematical Messenger.

The tumbler is a physical and mechanical phenomenon that people are very familiar with. The principle is that the objects that are light and heavy are relatively stable, that is, the lower the center of gravity, the more stable. When the tumbler is in equilibrium in the erect state, the distance between the center of gravity and the contact point is the smallest, that is, the center of gravity is the lowest. After deviating from the equilibrium position, the center of gravity will always rise. Therefore, the balance of the tumbler is a stable balance, and it always does not fall anyway.

Even if external forces change their balance, once the external force disappears, it will recover quickly. The tumbler has only one balance point and one non-equilibrium point, but the interior of the tumbler is uneven, and the lower part is heavy or has a counterweight attached to the bottom.

How can we find the function of having a tumbler, but the inside is uniform and has no weight; the mathematicians have racked their brains on this issue for many years. Hungarian mathematicians first made a more complex shape with plywood and placed them from different positions to see if they could automatically return to equilibrium after they fell over. The researchers found through mathematical calculations that any planar body has at least 2 equilibrium points and 2 unbalanced points. The researchers then experimented with the three-dimensional shape and made a unique shape from the physical model (see Figure 1). Its surface is a very complex rounded shape, and in nature, Indian star tortoises have a similar shape (Figure 2).

The researchers called the above-mentioned strange shape "Gomboc". It has the function of a tumbler completely, that is, there is only one balance point and one non-equilibrium point. After being tilted or tipped by an external force, once the external force disappears, it will automatically return to the original balance point.

Experts pointed out that making "Gomboc" into a toy will have great economic value. In 1974, the Hungarian Erno Rubik invented the Rubik's Cube toy, which was popular in the world. The invention was also published on the cover of the magazine Mathematical Messenger. (Reporter Dong Yingwei)

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