Imagine having a sandwich in front of you with two slices of bread and in the middle a slice of ham, which you want to cut with a single straight cut with a knife, so that the top bread, the bottom bread and the ham are divided exactly in half. This is something that makes us wonder: would it always be possible no matter how the ingredients are arranged? And this, which for our intuition has a relatively simple answer, for mathematicians has been the basis for creating the ‘ham sandwich theorem‘.
His story. Although the name sounds like a university hallway joke, we are dealing with a very serious classical mathematical theorem. For trace its originwe must travel to 1938, where there is the first known evidence of this problem and which appeared in a note by the Polish mathematician Hugo Steinhaus in the magazine Mathesis Polska. The difference is that at that time slices of bread were not used, but rather he simultaneously bisected meat, bone and fat from a ham with a single flat cut.
Although it must be detailed that Steinhaus proposed his conjecture, but it was the mathematician Stefan Banach who managed to prove it, making it exist for a long time. many doubts about who had the attribution of this theorem.
How it was demonstrated. To see that the perfect cut really exists, the mathematician had to resort to topology and reduce the problem to the so-called Borsuk-Ulam theorem. Without going into the formulas for this, Banach’s proof uses the unit sphere.
To give us an idea, if we consider all the possible directions in which we can point our knife on the sandwich, we can define a continuous mathematical function that evaluates what volume fraction of each ingredient remains on the “positive side” of the knife. Here the Borsuk-Ulam theorem guarantees us that, in any three-dimensional sphere, there are always two opposite points that will have the same result.
In summary. In practical terms, this means that mathematics assures us that there will always be an exact angle and position for the knife that will balance the volumes of bread, ham and cheese in an exact 50% ratio.
What is it for? Beyond the anecdotal aspect of seeing how a sandwich stars in a mathematical theorem, it is important to highlight that it has applications in geometry and computer science to create complete algorithms to be able to process a large amount of data.
Today, this problem is a classic in mathematics teaching. Dissemination platforms in Spanish, such as the well-known blog Gaussians or the channel Smyth Academyit they use usually to explain advanced topological concepts in an intuitive way, and all ultimately due to this concept that we have all done at some point in the kitchen.
Images | Suea Sivilaisith
In Xataka | This ranking shows which scientists have saved the most lives with their discoveries
GIPHY App Key not set. Please check settings