The strangely serious implications of the ‘Ham Sandwich Theorem’ in mathematics

Think about lunch. Possibly a pleasant ham sandwich. A slice of a knife cuts the ham and its two slices of bread neatly in half. However what if it slips? Oops, the ham is now folded beneath an overturned plate, with one piece of bread caught to the ground and the opposite to the ceiling. This is some comfort: geometry ensures {that a} single straight minimize, maybe utilizing a machete the scale of a home, can nonetheless completely bisect three slices of your rolled lunch, leaving precisely half of the ham and half of every slice of bread on both aspect. Reduce as a result of the “Ham Sandwich Theorem” in arithmetic guarantees that for any three (probably uneven) objects any orientation, there may be all the time some straight minimize that bisects all of them concurrently. This reality has some curious implications in addition to some thought-provoking factors because it pertains to gerrymandering in politics.

The theory generalizes to different dimensions as properly. One other mathematical expression says that n objects in n-dimensional house could be concurrently bisected by a (n – 1)-Dimensional Reduce. That ham sandwich is a little bit of a mouthful, however we’ll make it extra digestible. On a two-dimensional piece of paper, you’ll be able to draw something two form you need, and there’ll all the time be a (one-dimensional) straight line that cuts each completely in half. To ensure equal cuts for all three objects, we have to graduate in three dimensions and minimize with a two-dimensional airplane: consider that house-wrecking machete as sliding a skinny piece of paper between two elements of the home. In three dimensions, the machete has three levels of freedom: you’ll be able to scan it forwards and backwards throughout the room, then cease and rotate it to totally different angles and then Additionally Slice back and forth (as carrots are sometimes minimize diagonally, not straight).

For those who can think about a four-dimensional ham sandwich, as mathematicians love to do, you’ll be able to bisect a fourth element with a three-dimensional minimize.

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To get a style of methods to show the ham sandwich theorem, take into account a simplified model: two shapes in two dimensions the place one is a circle and the opposite is a blob. Each line that passes by way of the middle of a circle bisects it (asymmetrical shapes don’t essentially have such a middle; we at the moment are utilizing a circle to make our lives simpler). How do we all know that one in all these traces bisects the blob? Choose a line by way of the middle of the circle that does not intersect the blob in any respect. As proven within the first panel beneath, 100% of the blob is beneath the road. Now slowly rotate the road across the heart of the circle like a windmill. Lastly, it breaches the blob, cuts off extra of it, after which strikes beneath it the place the blob is beneath the zero % line. From this course of, we will infer that there should be a second the place precisely 50 % of the blob is beneath the road. We’re consistently going from 100% to zero %, so we should cross each amount in between, which means in some unspecified time in the future we’re precisely at 50 % (calculus followers might acknowledge this because the Intermediate Worth Theorem).

The animation illustrates the ham sandwich theorem using a circle, blob, and swirling line.  The line briefly turns red to highlight the moment it bisects both the circle and the blob.
Credit score: Amanda Montanez

This argument proves that there are traces that concurrently bisect our shapes (though it would not inform us the place that line is). It depends on the handy reality that each line by way of the middle of a circle bisects it, so we will freely rotate our line and deal with the blob with out worrying about neglecting the circle. For 2 uneven shapes we’d like a finer model of the windmill trick, and extensions to a few dimensions contain extra refined reasoning.

Apparently, the theory is true even when the ham and bread are damaged into a number of items. Use a cookie cutter to punch out a ham snowman and bake your bread into croutons; A superbly even minimize will all the time exist (not half of every snowman or crouton, however the complete quantity of ham and bread). Taking this concept to an excessive, we will make comparable claims about factors. Unfold your paper with pink and inexperienced dots, and there’ll all the time be a straight line that’s precisely half pink and half inexperienced. This model requires a small technicality: factors precisely on the dividing line could be counted on both aspect or not counted in any respect (for instance, when you have an odd quantity pink then you’ll be able to by no means divide them evenly with out this caveat).

Think about the unusual impact right here. You may draw a line throughout america in order that precisely half the nation has skunks And Half of its Twix bars are above the road. Whereas skunks and Twix bars aren’t really singular factors, they may as properly be when in comparison with the huge canvas of American landmass. Kicking issues up a notch, you’ll be able to draw a circle on the world (slicing a globe to go away a round cross-section) that incorporates half of the world is stone, half is paper, and half is scissors, or every other zany division you need.

As talked about, the ham sandwich theorem has far much less peculiar penalties for the perennial drawback of gerrymandering in politics. In america, state governments divide their states into electoral districts and every district elects one member to the Home of Representatives. Gerrymandering is the apply of intentionally drawing district boundaries for political achieve. For a simplified instance, think about a state with a inhabitants of 80 folks. Amongst them, 75 % (60 folks) favor the purple occasion, and 25 % (20 folks) desire the yellow occasion. The state will likely be divided into 4 districts of 20 folks every. It appears honest that three districts (75 %) ought to go purple and one other ought to go yellow in order that the state’s illustration in Congress conforms to the preferences of the inhabitants. Nevertheless, a artful cartographer can draw district boundaries so that every district has 15 purple-voters and 5 yellow-voters. Thus, purple would have a majority in each district and 100% of the state’s illustration would come from the purple occasion as an alternative of 75 %. Certainly, with sufficient voters, any The proportion that one aspect can use to win over the opposite (say 50.01 % purple vs. 49.99 % yellow) per district; Merely put, 50.01 % of every district helps the bulk occasion.

The graphic shows a field of colored dots with curved lines dividing the field so that each segment contains 15 purple dots and five yellow dots.
Credit score: Amanda Montanez

In fact these districts look very synthetic. The seemingly apparent approach is to impose restrictions on the scale of districts to cut back gerrymandering and deny the tentacled monsters we so typically see on American electoral maps. Certainly many states impose guidelines like this. Whereas it might appear that mandating districts to have a “regular” form would go a good distance towards eliminating the issue, intelligent Researchers utilized a sure geometric theorem to indicate that this can be a bunch of baloney. Let’s take a look at our instance once more: 80 voters of which 60 are purple-supporters and 20 are yellow-supporters. The Ham Sandwich Theorem tells us that irrespective of how they’re distributed, we will draw a straight line with precisely half purple voters and half yellow voters on both aspect (30 purple and 10 yellow on both aspect). Now deal with both sides of the minimize as its personal ham sandwich drawback, dividing every half by its personal straight line so that every ensuing area incorporates 15 purple and 5 yellow. Purple now has the identical gerrymandered benefit as earlier than (they received each district), however the ensuing districts are easier with straight-line boundaries!

The graphic shows a field of colored dots with straight lines dividing the field so that each segment contains 15 purple dots and five yellow dots.
Credit score: Amanda Montanez

Barber ham sandwich subdivisions will all the time create comparatively easy districts (mathematically talking they’re convex polygons the place they share a border with current state boundaries). Because of this primary rules on the scale of congressional districts can’t probably rule out even the worst examples of gerrymandering. Whereas math and politics might seem to be distant fields, an idle geometric diversion teaches us that essentially the most natural-sounding answer to gerrymandering would not minimize the mustard.

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