The ham sandwich theorem
Beyond the core curriculum — volume and solid geometry (a Polish accent, the Lviv school). Any three solids in space can be divided by a single flat cut into two equal-volume parts at once — regardless of their arrangement. The conjecture was posed by Hugo Steinhaus and proved by Stefan Banach (from the Borsuk–Ulam theorem), around 1938.
I’ll give you the number of objects and the dimension of the space — decide whether one cut is sure to bisect them all at once.
What just happened?
You changed the arrangement of the three objects — from a neat sandwich to completely scattered ingredients. Each time one line (in space: one plane) divided the bread, cheese and ham simultaneously into two equal “volume” halves. You did not have to arrange anything neatly — such a cut always exists.
The ham sandwich theorem:
It is one of the most surprising and amusingly named results in topology — with an important Polish contribution. It says that any three solids placed in three-dimensional space — picture a slice of bread, a slice of cheese and a slice of ham, arranged any way at all — can be divided by a single straight cut (one plane) so that each of the three ingredients is bisected into two equal-volume parts.
In general (Stone–Tukey, 1942):
objects in -dimensional space can be bisected by one hyperplane (dividing the space into two parts and ):
For (bread, cheese, ham) the hyperplane is an ordinary plane, and the “volume” is the volume of each of the three solids.
How do we know? (and why it works in the model)
The conjecture was posed by Hugo Steinhaus (a co-founder of the Lviv school of mathematics), and it was proved by Stefan Banach using the Borsuk–Ulam theorem (due to the Polish mathematician Karol Borsuk). In our illustrative 2D model each object is a centrally symmetric figure (an ellipse), and a line bisects such a figure exactly when it passes through its centre — which is why a single line running through the centres of all three objects bisects them simultaneously.
Why it is beautiful:
It is a model example of how abstract topology (fixed-point and antipodal theorems) gives a very concrete, “kitchen” conclusion — and how Polish mathematicians of the interwar period helped shape world mathematics.
Remember: any three solids in space can be bisected at once by one plane — such a cut always exists. This is the ham sandwich theorem (beyond the core curriculum — volume and solid geometry; a Polish accent: the Lviv school).
Theory & formulas
Statement
Any three solids in space (bread, cheese, ham) can be bisected by a single flat cut simultaneously by volume — there is a plane dividing each of them into two equal parts, regardless of their arrangement.
Where it comes from
It follows from the Borsuk–Ulam theorem: on every sphere there are two antipodal points at which a continuous function has equal values. "The volume on one side" is such a function of the direction of the cut.
The Polish contribution
The conjecture was posed by Hugo Steinhaus, and the proof was given by Stefan Banach (the Lviv school, around 1938).
Frequently asked questions
›Can one cut bisect three arbitrarily arranged solids?
Yes — the ham sandwich theorem guarantees the existence of a plane halving all three at once.
›Where does the name "ham sandwich" come from?
From the image: bread, cheese and ham (three solids) can be bisected simultaneously by a single stroke of the knife.
›What is the Polish contribution to this theorem?
The conjecture was posed by Hugo Steinhaus, and Stefan Banach proved it (the Lviv school), using the Borsuk–Ulam theorem.
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