The Strangely Serious Implications of Math’s ‘Ham Sandwich Theorem’

Think about lunch. Maybe a pleasant ham sandwich. A slice of a knife neatly halves the ham and its two bread slices. However what in the event you slip? Oops, the ham now rests folded below a flipped plate, with one slice of bread on the ground and the opposite caught to the ceiling. Right here’s some solace: geometry ensures {that a} single straight minimize, 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 minimize. That’s as a result of math’s “ham sandwich theorem” guarantees that for any three (probably uneven) objects in any orientation, there’s all the time some straight minimize 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 theory 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 minimize. 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 may draw no matter two shapes you need, and there’ll all the time be a (one-dimensional) straight line that cuts each completely in half. To ensure an equal minimize for 3 objects, we have to graduate to a few dimensions and minimize them with a two-dimensional airplane: 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 may scan it backwards and forwards throughout the room, then cease and rotate it to totally different angles, and then additionally rock the machete backward and forward (like how carrots are sometimes minimize obliquely, and never straight).

In the event you can think about a four-dimensional ham sandwich, as mathematicians love to do, then you would additionally bisect a fourth ingredient with a three-dimensional minimize.

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To get a taste for methods 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. Each line that passes by way of 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 strains additionally bisects the blob? Decide a line by way of the middle of the circle that doesn’t intersect the blob in any respect. As depicted within the first panel beneath, one hundred pc of the blob lies beneath the road. Now slowly rotate the road across the middle of the circle like a windmill. Finally, it breaches the blob, cuts by way of an increasing number of of it, after which passes beneath it the place zero p.c of the blob lies beneath the road. From this course of, we will deduce that there should be a second at which precisely 50 p.c of the blob lies beneath the road. We’re steadily transferring from one hundred pc down repeatedly to zero p.c, so we should cross each quantity in between, that means sooner or later we’re at precisely 50 p.c (calculus fans would possibly acknowledge this because the intermediate worth theorem).

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 by way of the middle of a circle bisects it, so we may freely rotate our line and give attention to 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 subtle arguments.

Apparently 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 minimize will all the time exist (every snowman or crouton received’t essentially be halved, however the complete quantity of ham and bread will probably be). Taking this concept to its excessive, we will make an identical declare about factors. Scatter your paper with purple and inexperienced dots, and there’ll all the time 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, if in case you have an odd number of reds then you would by no means break up them evenly with out this caveat).

Ponder the weird implications right here. You’ll be able to 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. Though skunks and Twix bars should not really single factors, they could as properly be when in comparison with the huge canvas of American landmass. Kicking issues up a dimension, you may draw a circle on Earth (slicing by way of 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 every other zany classes you would like.

As talked about, the ham sandwich theorem carries far much less whimsical penalties for the perennial downside of gerrymandering in politics. Within the U.S., state governments divide their states into electoral districts, and every district elects a member to the Home of Representatives. Gerrymandering is the apply of carving out these district boundaries intentionally for political acquire. For a simplified instance, think about a state with a inhabitants of 80 individuals. 75 p.c of them (60 individuals) favor the purple celebration, and 25 p.c (20 individuals) want the yellow celebration. The state will probably be divided into 4 districts of 20 individuals every. It appears honest that three of these districts (75 p.c) 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. Nevertheless, a artful cartographer may squiggle district boundaries in such a manner that every district incorporates 15 purple-voters and 5 yellow-voters. This fashion, purple would maintain a majority in each district and one hundred pc of the state’s illustration would come from the purple celebration relatively than 75 p.c. The truth is, with sufficiently many citizens, any proportion edge that one celebration has over one other (say 50.01 p.c purple vs. 49.99 p.c yellow) might be exploited to win each district; simply make it so 50.01 p.c of each district helps the bulk celebration.

In fact these districts look extremely synthetic. A seemingly apparent technique to curtail gerrymandering could be to position restrictions on the shapes of the districts and disallow the tentacled monstrosities that we frequently see on American electoral maps. Certainly many states impose guidelines like this. Whereas it’d appear to be mandating districts to have “regular” shapes would go a good distance in ameliorating the issue, clever researchers have utilized a sure geometric theorem to point out 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 each side). Now deal with both sides of the minimize as its personal ham sandwich downside, 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!

Repeated ham sandwich subdivision will all the time produce comparatively easy districts (in math-speak they’re convex polygons besides the place they probably share a boundary with an present state border). Which means that fundamental rules on the shapes of congressional districts most likely can’t preclude even the worst situations of gerrymandering. Though math and politics might appear to be distant fields, an idle geometric diversion taught us that essentially the most natural-sounding resolution to gerrymandering doesn’t minimize the mustard.