Home » The Surprisingly Critical Implications of Math’s ‘Ham Sandwich Theorem’

The Surprisingly Critical Implications of Math’s ‘Ham Sandwich Theorem’

by Green Zak
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Consider lunch. Perhaps a pleasant ham sandwich. A slice of a knife neatly halves the ham and its two bread slices. But what in case you slip? Oops, the ham now rests folded underneath a flipped plate, with one slice of bread on the ground and the opposite caught to the ceiling. Here’s some solace: geometry ensures {that a} single straight reduce, maybe utilizing a room-sized machete, can nonetheless completely bisect all three items of your tumbled lunch, leaving precisely half of the ham and half of every slice of bread on both aspect of the reduce. That’s as a result of math’s “ham sandwich theorem” guarantees that for any three (probably uneven) objects in any orientation, there’s at all times some straight reduce that concurrently bisects all of them. This reality has some weird implications in addition to some sobering ones because it pertains to gerrymandering in politics.

The theorem generalizes to different dimensions as properly. A extra mathematical phrasing says that n objects in n-dimensional area might be concurrently bisected by an (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 possibly can draw no matter two shapes you need, and there’ll at all times be a (one-dimensional) straight line that cuts each completely in half. To assure an equal reduce for 3 objects, we have to graduate to a few dimensions and reduce them with a two-dimensional aircraft: consider that room-ravaging machete as slipping a skinny piece of paper between the 2 halves of the room. In three dimensions, the machete has three levels of freedom: you possibly can scan it backwards and forwards throughout the room, then cease and rotate it to completely different angles, and then additionally rock the machete back and forth (like how carrots are sometimes reduce obliquely, and never straight).

If you possibly can think about a four-dimensional ham sandwich, as mathematicians love to do, then you could possibly additionally bisect a fourth ingredient with a three-dimensional reduce.

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To get a taste for find out how to show the ham sandwich theorem, think about a simplified model: two shapes in two dimensions the place one among them is a circle and the opposite is a blob. Every line that passes via the middle of a circle bisects it (asymmetrical shapes don’t essentially have a middle like this; we’re utilizing a circle to make our lives simpler for now). How do we all know that one among these traces additionally bisects the blob? Pick a line via the middle of the circle that doesn’t intersect the blob in any respect. As depicted within the first panel under, 100% of the blob lies under the road. Now slowly rotate the road across the heart of the circle like a windmill. Eventually, it breaches the blob, cuts via increasingly of it, after which passes under it the place zero % of the blob lies under the road. From this course of, we will deduce that there should be a second at which precisely 50 % of the blob lies under the road. We’re steadily shifting from 100% down repeatedly to zero %, so we should cross each quantity in between, that means sooner or later we’re at precisely 50 % (calculus followers would possibly acknowledge this because the intermediate worth theorem).

Animation illustrates the ham sandwich theorem using a circle, blob and rotating line. The line briefly turns red to highlight the moment when it bisects both the circle and the blob.
Credit: Amanda Montañez

This argument proves that there’s some line that concurrently bisects our shapes (though it doesn’t inform us the place that line is). It depends on the handy reality that each line via the middle of a circle bisects it, so we might freely rotate our line and concentrate on the blob with out worrying about neglecting the circle. Two uneven shapes require a subtler model of our windmill approach, and the extension to a few dimensions entails extra refined arguments.

Interestingly the theory holds true even when the ham and bread are damaged into a number of items. Use a cookie cutter to punch out ham snowmen, and bake your bread into croutons; a wonderfully equal reduce will at all times exist (every snowman or crouton received’t essentially be halved, however the complete quantity of ham and bread will likely be). Taking this concept to its excessive, we will make an identical declare about factors. Scatter your paper with crimson and inexperienced dots, and there’ll at all times be a straight line with precisely half of the reds and half of the greens on both aspect of it. This model requires a small technicality: factors that lie precisely on the dividing line might be counted on both aspect or not counted in any respect (for instance, in case you have an odd quantity of reds then you could possibly by no means break up them evenly with out this caveat).

Contemplate the weird implications right here. You can draw a line throughout the U.S. in order that precisely half of the nation’s skunks and half of its Twix bars lie above the road. Although skunks and Twix bars will not be really single factors, they may as properly be when in comparison with the huge canvas of American landmass. Kicking issues up a dimension, you possibly can draw a circle on Earth (slicing via a globe leaves a round cross part) that incorporates half of the world’s rocks, half of its paper, and half of its scissors, or another zany classes you want.

As talked about, the ham sandwich theorem carries far much less whimsical penalties for the perennial drawback of gerrymandering in politics. In the U.S., state governments divide their states into electoral districts, and every district elects a member to the House of Representatives. Gerrymandering is the apply of carving out these district boundaries intentionally for political achieve. For a simplified instance, think about a state with a inhabitants of 80 individuals. 75 % of them (60 individuals) favor the purple social gathering, and 25 % (20 individuals) choose the yellow social gathering. The state will likely be divided into 4 districts of 20 individuals every. It appears truthful that three of these districts (75 %) ought to go to purple and the opposite one ought to go to yellow in order that the state’s illustration in Congress accords with the preferences of the inhabitants. However, a artful cartographer might squiggle district boundaries in such a method that every district incorporates 15 purple-voters and 5 yellow-voters. This method, purple would maintain a majority in each district and 100% of the state’s illustration would come from the purple social gathering reasonably than 75 %. In reality, with sufficiently many citizens, any proportion edge that one social gathering has over one other (say 50.01 % purple vs. 49.99 % yellow) might be exploited to win each district; simply make it so 50.01 % of each district helps the bulk social gathering.

Graphic shows a field of colored dots with tortuously curved lines dividing the field such that each section contains 15 purple dots and five yellow dots.
Credit: Amanda Montañez

Of course these districts look extremely synthetic. A seemingly apparent approach to curtail gerrymandering can be to position restrictions on the shapes of the districts and disallow the tentacled monstrosities that we regularly see on American electoral maps. Indeed many states impose guidelines like this. While it’d seem to be mandating districts to have “regular” shapes would go a great distance in ameliorating the issue, intelligent researchers have utilized a sure geometric theorem to indicate how that’s a bunch of baloney. Let’s revisit our instance: 80 voters comprising 60 purple-supporters and 20 yellow-supporters. The ham sandwich theorem tells us that regardless of how they’re distributed, we will draw a straight line with precisely half of the purple voters and half of the yellow voters on both aspect (30 purple and 10 yellow on either side). Now deal with both sides of the reduce as its personal ham sandwich drawback, splitting every half with their very own straight line so that each ensuing area incorporates 15 purples and 5 yellows. Purple now has the identical gerrymandered benefit as earlier than (they win each district), however the ensuing areas are all easy with straight-line boundaries!

Graphic shows a field of colored dots with straight lines dividing the field such that each section contains 15 purple dots and five yellow dots.
Credit: Amanda Montañez

Repeated ham sandwich subdivision will at all times produce comparatively easy districts (in math-speak they’re convex polygons besides the place they probably share a boundary with an current state border). This signifies that fundamental laws on the shapes of congressional districts most likely can’t preclude even the worst cases of gerrymandering. Although math and politics could seem to be distant fields, an idle geometric diversion taught us that essentially the most natural-sounding answer to gerrymandering doesn’t reduce the mustard.

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