Showing posts with label visualization. Show all posts
Showing posts with label visualization. Show all posts
30 June 2010
How not to visualize the electoral college
I went to the National Constitution Center in Philadelphia today.
As you may know, the Constitution provides that, in elections for the President, each state receives a number of electors equal to its total number of senators and representatives. Each state has two senators, and the number of representatives is proportional to the population. The number of representatives is adjusted after the census, which happens in years divisible by ten.
Why am I telling you this? Because at one point on the wall there was an animated map, which displayed how apportionment had changed between censuses. Each state was represented as a "cylinder", with base the state itself and height proportional to its number of electors. (Or representatives; it honestly would be impossible to tell the difference by eye, as in this scheme that would just push everything up by two units.) There was one such display in the animation for each census, with smooth transitions between them.
Since the eye wants to interpret the "volume" of a state as its number of electors, this has the effect of making geographically-large states look like they have better representation than they do. I noticed this by looking at New Jersey and Pennsylvania, which have areas of 7417 and 44817 square miles, and 15 and 21 electors respectively. The solid corresponding to Pennsylvania has about eight times the volume as that corresponding to New Jersey. New Jersey's an easy one to look at because it happens to be the most densely populated state at the present time, and in this visualization it is not the tallest.
The volume of the solid corresponding to each state is proportional to the product of its number of its electors and its area. The states for which this product is largest are, in order, Texas, California, Alaska, New York, Florida, Illinois, Arizona, Michigan, Pennsylvania, and Colorado. The first two of these, between them, have 41% of the total volume in this visualization.
I'd suggest replacing this with a model where volume is proportional to the number of electoral votes. Or, since that might have its own problems, a cartogram which evolves in time. The West would just grow out of nowhere.
As you may know, the Constitution provides that, in elections for the President, each state receives a number of electors equal to its total number of senators and representatives. Each state has two senators, and the number of representatives is proportional to the population. The number of representatives is adjusted after the census, which happens in years divisible by ten.
Why am I telling you this? Because at one point on the wall there was an animated map, which displayed how apportionment had changed between censuses. Each state was represented as a "cylinder", with base the state itself and height proportional to its number of electors. (Or representatives; it honestly would be impossible to tell the difference by eye, as in this scheme that would just push everything up by two units.) There was one such display in the animation for each census, with smooth transitions between them.
Since the eye wants to interpret the "volume" of a state as its number of electors, this has the effect of making geographically-large states look like they have better representation than they do. I noticed this by looking at New Jersey and Pennsylvania, which have areas of 7417 and 44817 square miles, and 15 and 21 electors respectively. The solid corresponding to Pennsylvania has about eight times the volume as that corresponding to New Jersey. New Jersey's an easy one to look at because it happens to be the most densely populated state at the present time, and in this visualization it is not the tallest.
The volume of the solid corresponding to each state is proportional to the product of its number of its electors and its area. The states for which this product is largest are, in order, Texas, California, Alaska, New York, Florida, Illinois, Arizona, Michigan, Pennsylvania, and Colorado. The first two of these, between them, have 41% of the total volume in this visualization.
I'd suggest replacing this with a model where volume is proportional to the number of electoral votes. Or, since that might have its own problems, a cartogram which evolves in time. The West would just grow out of nowhere.
25 August 2008
Sign errors and upside-down things
When was the last time you made a sign error?Marcello Herreshoff writes at Overcoming Bias on using the native architecture, i. e. taking advantage of the things that the human brain is naturally optimized for when trying to do more recently invented things, like mathematics.
When was the last time you visualized something upside-down by mistake?
I thought so.
(Although I did visualize something upside-down today. In particular, I drew the Hasse diagram of a certain poset and then realized that the way I was thinking of things, I really wanted the reverse order. But that doesn't really count, because I'd never seen this particular poset before.)
26 July 2008
Bill Rankin's population density graphs
Last week I wrote a post about population densities.
Take a look at the interesting graphs at Bill Rankin's Radical Cartography; they show how population density is related to:
Take a look at the interesting graphs at Bill Rankin's Radical Cartography; they show how population density is related to:
- racial and ethnic groups (American Indians and Alaska Natives, not surprisingly, live at the lowest population densities; what surprised me was the large amount of Hispanic population at between 1 and 10 per square mile, which Rankin says might correspond to ranchers);
- age. Roughly speaking, people ages 18 to 39 or under 5 are overrepresented at "high" densities (above 4000 or so), and other ages are overrepresented at "low" densities (below that same cutoff). This is, I suspect, a reflection of people moving to the city when they leave their parents house, and then leaving the city when it's time for their kids to go to school.
- income is highest at suburban and central-city densities, with a valley in between. Not surprising; in general the central part of a city is rich, it's surrounded by poorer neighborhoods, and then eventually income starts going up again. Rural places are poor as well.
- gender -- there are more women at high density, which I can't explain.
- population and area -- I tried to make a plot like this but had some trouble, because I was just playing around with output from another web site and didn't have the raw data.
07 June 2008
Virtual laboratories in probability and statistics
Virtual laboratories in probability and statistics offers several dozen applets for the visualization of probabilistic systems.
Most of them are pretty simple -- they graph some theoretical distribution and then show the results of repeating some corresponding experiment that gives that over and over again.
For example, the random walk experiment -- although I link to that one because it shows the arcsine distribution for a random walk quite well! The "arcsine distribution" as applied to random walks is as follows: let X1, X2, ... be independent random variables which are each +1 with probability 1/2 and -1 with probability 1/2. Let Sn = X1 + X2 + ... + Xn. Consider the sequence (S1, ..., S2n) -- now, what's the probability that the last zero in this sequence comes between time 2an and 2bn? It turns out that as n approaches infinity, this probability approaches
and in particular, the last zero is surprisingly likely to be near 0 or 2n, and surprisingly unlikely to be somewhere in the middle.
If you like pretty pictures (and you should -- a picture is worth a thousand words and all that), check out the section on interacting particle systems, which includes the fire process (which models the spread of a fire in a forest, where the trees form a grid and fires only spread between adjacent trees in discrete time -- so a bit unrealistic as a model for real fires, but who cares?) and the voter process. Here h particles in a grid change state in order to match their neighbors; "segregation" of the different particle types occurs, not because anybody's enforcing it from on high but just because of these local constraints, and some particle types just die out completely.
The web page not only includes the applets, but explanations of what's going on, since the intended audience here seems to be students taking a first course in probability.
Most of them are pretty simple -- they graph some theoretical distribution and then show the results of repeating some corresponding experiment that gives that over and over again.
For example, the random walk experiment -- although I link to that one because it shows the arcsine distribution for a random walk quite well! The "arcsine distribution" as applied to random walks is as follows: let X1, X2, ... be independent random variables which are each +1 with probability 1/2 and -1 with probability 1/2. Let Sn = X1 + X2 + ... + Xn. Consider the sequence (S1, ..., S2n) -- now, what's the probability that the last zero in this sequence comes between time 2an and 2bn? It turns out that as n approaches infinity, this probability approaches
and in particular, the last zero is surprisingly likely to be near 0 or 2n, and surprisingly unlikely to be somewhere in the middle.
If you like pretty pictures (and you should -- a picture is worth a thousand words and all that), check out the section on interacting particle systems, which includes the fire process (which models the spread of a fire in a forest, where the trees form a grid and fires only spread between adjacent trees in discrete time -- so a bit unrealistic as a model for real fires, but who cares?) and the voter process. Here h particles in a grid change state in order to match their neighbors; "segregation" of the different particle types occurs, not because anybody's enforcing it from on high but just because of these local constraints, and some particle types just die out completely.
The web page not only includes the applets, but explanations of what's going on, since the intended audience here seems to be students taking a first course in probability.
18 April 2008
Visualizing the Sieve of Eratosthenes (in the May Notices of the AMS)
From the May Notices of the AMS: Visualizing the Sieve of Eratosthenes, by David N. Cox.
The basic idea of the article is that we can color the point (n, m) whenever m and n are positive integers and m divides n; then interesting patterns appear among lattice points in the first quadrant. Alternatively, the row y = n contains colored points at x = n, 2n, 3n, ... and in general every nth column. The procedure generalizes the sieve of Erastosthenes; the number of marked points in the column x = n is just the number of divisors of n.
One pattern that appears comes about when we mark (3,1), (6,2), (9,3), and so on, so all the lattice points on a diagonal through the origin of slope 1/3 are colored; something similar happens for every k. But diagonals also appear "radiating" from points on the x-axis which are not the origin. For example, radiating out of the point (2520, 0) we have a diagonal of slope 1, which passes through the marked points (2521, 1), (2522, 2), ..., (2530, 10); this occurs because 2520 is divisible by each of 1 through 10. In general one sees
Cox points out some mysterious-seeming parabolic patterns of the marked points; for example he mentions a left-opening parabola with vertex (17956, 134). Now, 1342 is 17956 -- and so the parabola constains the points (17956-x2, 134 ± x) for each x. In fact, we have a parabola containing the points (k2-x2, k ± x) for each integer k. Cox says that no right-opening parabolas are observed; these would correspond to factorizations of the form (k2+x2, k ± ix) where i is the imaginary unit, but he's working over the integers! Of course, if you stare at the points you might see what you think are right-opening parabolas, but the appearance of those is probably just a coincidence. At this point I will wave my hands and intone the magic words "Ramsey theory". In any case, right-opening parabolas, should they exist, are certainly not given by such a simple rule.
(In the interests of the sentence preceding this one being entirely correct, I will define "simple" as "things I thought of while writing this post".)
The basic idea of the article is that we can color the point (n, m) whenever m and n are positive integers and m divides n; then interesting patterns appear among lattice points in the first quadrant. Alternatively, the row y = n contains colored points at x = n, 2n, 3n, ... and in general every nth column. The procedure generalizes the sieve of Erastosthenes; the number of marked points in the column x = n is just the number of divisors of n.
One pattern that appears comes about when we mark (3,1), (6,2), (9,3), and so on, so all the lattice points on a diagonal through the origin of slope 1/3 are colored; something similar happens for every k. But diagonals also appear "radiating" from points on the x-axis which are not the origin. For example, radiating out of the point (2520, 0) we have a diagonal of slope 1, which passes through the marked points (2521, 1), (2522, 2), ..., (2530, 10); this occurs because 2520 is divisible by each of 1 through 10. In general one sees
Cox points out some mysterious-seeming parabolic patterns of the marked points; for example he mentions a left-opening parabola with vertex (17956, 134). Now, 1342 is 17956 -- and so the parabola constains the points (17956-x2, 134 ± x) for each x. In fact, we have a parabola containing the points (k2-x2, k ± x) for each integer k. Cox says that no right-opening parabolas are observed; these would correspond to factorizations of the form (k2+x2, k ± ix) where i is the imaginary unit, but he's working over the integers! Of course, if you stare at the points you might see what you think are right-opening parabolas, but the appearance of those is probably just a coincidence. At this point I will wave my hands and intone the magic words "Ramsey theory". In any case, right-opening parabolas, should they exist, are certainly not given by such a simple rule.
(In the interests of the sentence preceding this one being entirely correct, I will define "simple" as "things I thought of while writing this post".)
14 February 2008
Pretty pictures!
From MathTrek: Math on Display. Apparently there was an exhibition of mathematical art at the Joint Mathematics Meetings in San Diego last month. Some of it actually arises naturally in considering problems which actually arise. Since I mostly deal in discrete mathematics I don't see the really nice pictures too often in my own work -- they seem to arise more naturally on the continuous side of things -- although certainly there are a lot of results in discrete mathematics that are essentially limit theorems, which often give rise to nice "smooth" pictures.
Although it doesn't arise from a mathematical problem, my favorite of the pictures shown in that post is the picture of Sierpinski made out of tiles which are different iterations of the Sierpinski carpet. There's something deliciously self-referential about it. Now if only there were some set of pictures commonly associated with, say, Gödel...
Although it doesn't arise from a mathematical problem, my favorite of the pictures shown in that post is the picture of Sierpinski made out of tiles which are different iterations of the Sierpinski carpet. There's something deliciously self-referential about it. Now if only there were some set of pictures commonly associated with, say, Gödel...
16 November 2007
Moebius transformations revealed
Moebius transformations revealed, from YouTube, via MathTrek -- a visual demonstration of the folk theorem that Möbius transformations correspond to moving around the Riemann sphere in various ways. By Douglas Arnold and Jonathan Rogness.
27 September 2007
Fun with fugues
At Science after Sunclipse, Blake Stacey links to some videos from Stephen Malinowski's Music Animation Machine. The Music Animation Machine animates music; the basic idea is that notes are represented by rectangles, the length of which corresponds to the length of the note. What I found particularly interesting was that the videos use color to represent either:
Johann Sebastian Bach, Toccata and Fugue in D Minor:
[埋込みオブジェクト:http://www.youtube.com/v/ipzR9bhei_o]
Frederic Chopin, Etude, opus 10 #7:
[埋込みオブジェクト:http://www.youtube.com/v/vMyg9CJxWNA]
The harmonic coloration is based on the circle of fifths, which is an interesting solution to the problem that notes which are close together in pitch are not close together in some sort of "harmonic space".
I find myself wondering if the coloring based on voice could be automated. This would be trivial for things written entirely in, say, standard four-part voice leading as taught in an introductory music theory class, because there are always exactly four notes at any given time and the voices don't cross; it would be very nontrivial for actual music. (This isn't just a musical problem, believe it or not. A related problem is as follows: a baseball team has five starting pitchers, which it uses in a pitching rotation: ideally pitcher n pitches on days n, n+5, n+10, ... But there are off days, people get hurt, and so on. How do you decide when one pitcher has "replaced" another in the rotation? The people at Baseball Prospectus have thought about this -- sorry I can't find the link -- but their solutions basically involve just staring at a list of who pitched what day and writing numbers next to them kind of arbitrarily. It's not quite the same thing, though, because there's only one starting pitcher per game (relief pitchers are used in a much more ad hoc manner) but notes are played simultaneously. I suspect there are other problems of this sort where there are logical ways to sort some sequence of things into bins -- musical voices, rotation slots, and so on -- but none come to mind.)
The sort of notation they're using here seems like a logical historical antecedent of present-day musical notation (though I don't know enough about the history to know if it is); the main differences are that in modern notation we decide that seven of the notes in each octave are more important than the other five (and enshrine this in the notation) and we don't write a whole note as being, say, sixteen times as long as a sixteenth note. This is probably a good thing from the point of view of readability. It also resembles a player piano roll (modulo coloring) which I doubt is a coincidence.
Oh, and you have to watch the "Oops, I Did It Again" fugue, which I found from Good Math, Bad Math:
[埋込みオブジェクト:http://www.youtube.com/v/tgDcC2LOJhQ]
- the "voices" in the score, as in the first video below (Bach), and
- harmonic information, as in the second video below (Chopin)
Johann Sebastian Bach, Toccata and Fugue in D Minor:
[埋込みオブジェクト:http://www.youtube.com/v/ipzR9bhei_o]
Frederic Chopin, Etude, opus 10 #7:
[埋込みオブジェクト:http://www.youtube.com/v/vMyg9CJxWNA]
The harmonic coloration is based on the circle of fifths, which is an interesting solution to the problem that notes which are close together in pitch are not close together in some sort of "harmonic space".
I find myself wondering if the coloring based on voice could be automated. This would be trivial for things written entirely in, say, standard four-part voice leading as taught in an introductory music theory class, because there are always exactly four notes at any given time and the voices don't cross; it would be very nontrivial for actual music. (This isn't just a musical problem, believe it or not. A related problem is as follows: a baseball team has five starting pitchers, which it uses in a pitching rotation: ideally pitcher n pitches on days n, n+5, n+10, ... But there are off days, people get hurt, and so on. How do you decide when one pitcher has "replaced" another in the rotation? The people at Baseball Prospectus have thought about this -- sorry I can't find the link -- but their solutions basically involve just staring at a list of who pitched what day and writing numbers next to them kind of arbitrarily. It's not quite the same thing, though, because there's only one starting pitcher per game (relief pitchers are used in a much more ad hoc manner) but notes are played simultaneously. I suspect there are other problems of this sort where there are logical ways to sort some sequence of things into bins -- musical voices, rotation slots, and so on -- but none come to mind.)
The sort of notation they're using here seems like a logical historical antecedent of present-day musical notation (though I don't know enough about the history to know if it is); the main differences are that in modern notation we decide that seven of the notes in each octave are more important than the other five (and enshrine this in the notation) and we don't write a whole note as being, say, sixteen times as long as a sixteenth note. This is probably a good thing from the point of view of readability. It also resembles a player piano roll (modulo coloring) which I doubt is a coincidence.
Oh, and you have to watch the "Oops, I Did It Again" fugue, which I found from Good Math, Bad Math:
[埋込みオブジェクト:http://www.youtube.com/v/tgDcC2LOJhQ]
18 September 2007
A cultural map of the world
Take a look at the Inglehart-Welzel Cultural Map of the World:
The resulting "map" doesn't resemble any "traditional" map of the world, but it's interesting. For example, all the English-speaking countries end up near each other, all the countries of Protestant Europe end up near each other, and so on.
I'm not sure why "English-speaking" is its own group, while no other language is given its own group. In particular, all the Spanish-and-Portuguese-speaking countries seem to end up together. (This is obscured by the map, which for some strange reason includes Uruguay in "Catholic Europe" and Portugal in "Latin America". I'm conflating Spanish and Portuguese not because I don't know there's a difference, but because they are fairly similar as languages go.
My instinct is that the "Traditional Values"/"Secular-Rational Values" divide is similar to the ideological conservative/liberal divide in American politics (although I'm not sure how this would be made precise); I want to say that the "Survival Values"/"Self-Expression Values" dimension corresponds to the economic conservative/liberal divide in American politics although that seems like a lot more of a stretch. Apparently countries seem to move from "Survival" to "Self-Expression" (i. e. rightward on the graph) with time.
One sees a lot of these "factor analysis" plots where a large-dimensional space is reduced to just two dimensions in this way; I don't think there's some fundamental reason why two dimensions is the natural way to think about this, but rather that we're just good at drawing two-dimensional pictures. Dave Rusin's Mathematical Atlas includes such a plot (although naming the dimensions is tricky -- I thought they might be discrete vs. continuous and pure vs. applied.) I've also seen a political map like this, based on the voting records of U. S. Senators and Representatives; it's kind of fascinating to watch how the two parties have moved around with time, and how you can explain almost as much variation between politicians by just looking at a single variable as you used to need two variables for. You can almost predict who will vote for a given bill just by lining up the Senators from "most liberal" to "most conservative" and drawing a line somewhere to separate the two sides. Life is more complicated than that.
I wonder if one could use a plot like this (or the data which underlies it) to predict which international borders are likely to create a lot of tension. For example, the U. S. is (according to this plot) much more similar to Canada than it is to Mexico, and there seems to be a lot more tension at the U. S.'s southern border than at its northern one. Perhaps one could predict strife within a country as well, if a survey like this was done for subnational entities. Lumping the entire United States together seems almost ludicrous to me.
Also, how is this sort of thing correlated with language? And if the language spoken in a country changes, for whatever reason, is this correlated with that country becoming more like countries that speak the new language? It seems reasonable that there should be some connection between language and culture, if only because most of culture is expressed through language. But causation is a problem; do countries become more like each other because they speak the same language, or do countries that speak the same language become more like each other?
The World Values Surveys were designed to provide a comprehensive measurement of all major areas of human concern, from religion to politics to economic and social life and two dimensions dominate the picture: (1) Traditional/ Secular-rational and (2) Survival/Self-expression values. These two dimensions explain more than 70 percent of the cross-national variance in a factor analysis of ten indicators-and each of these dimensions is strongly correlated with scores of other important orientations.
The resulting "map" doesn't resemble any "traditional" map of the world, but it's interesting. For example, all the English-speaking countries end up near each other, all the countries of Protestant Europe end up near each other, and so on.
I'm not sure why "English-speaking" is its own group, while no other language is given its own group. In particular, all the Spanish-and-Portuguese-speaking countries seem to end up together. (This is obscured by the map, which for some strange reason includes Uruguay in "Catholic Europe" and Portugal in "Latin America". I'm conflating Spanish and Portuguese not because I don't know there's a difference, but because they are fairly similar as languages go.
My instinct is that the "Traditional Values"/"Secular-Rational Values" divide is similar to the ideological conservative/liberal divide in American politics (although I'm not sure how this would be made precise); I want to say that the "Survival Values"/"Self-Expression Values" dimension corresponds to the economic conservative/liberal divide in American politics although that seems like a lot more of a stretch. Apparently countries seem to move from "Survival" to "Self-Expression" (i. e. rightward on the graph) with time.
One sees a lot of these "factor analysis" plots where a large-dimensional space is reduced to just two dimensions in this way; I don't think there's some fundamental reason why two dimensions is the natural way to think about this, but rather that we're just good at drawing two-dimensional pictures. Dave Rusin's Mathematical Atlas includes such a plot (although naming the dimensions is tricky -- I thought they might be discrete vs. continuous and pure vs. applied.) I've also seen a political map like this, based on the voting records of U. S. Senators and Representatives; it's kind of fascinating to watch how the two parties have moved around with time, and how you can explain almost as much variation between politicians by just looking at a single variable as you used to need two variables for. You can almost predict who will vote for a given bill just by lining up the Senators from "most liberal" to "most conservative" and drawing a line somewhere to separate the two sides. Life is more complicated than that.
I wonder if one could use a plot like this (or the data which underlies it) to predict which international borders are likely to create a lot of tension. For example, the U. S. is (according to this plot) much more similar to Canada than it is to Mexico, and there seems to be a lot more tension at the U. S.'s southern border than at its northern one. Perhaps one could predict strife within a country as well, if a survey like this was done for subnational entities. Lumping the entire United States together seems almost ludicrous to me.
Also, how is this sort of thing correlated with language? And if the language spoken in a country changes, for whatever reason, is this correlated with that country becoming more like countries that speak the new language? It seems reasonable that there should be some connection between language and culture, if only because most of culture is expressed through language. But causation is a problem; do countries become more like each other because they speak the same language, or do countries that speak the same language become more like each other?
15 August 2007
visualizing higher dimensions
Jon Shock writes at SciTalks about visualizing things in four dimensions, although it's actually less about visualization than about how we don't need to visualize so long as we can calculate -- although he does provide a link to a youtube video showing how one can obtain the projection of a hypercube onto a plane.
I've never been good at visualizing things in more than three dimensions (actually, to tell the truth, I have trouble visualizing things in my head in more than two dimensions; to visualize anything successfully in three dimensions I have to use my hands and point at things in the empty space in front of me). But a guy I knew in college claimed to be able to visualize things in five dimensions fairly easily. The fourth dimension was time -- so a four-dimensional sphere, for example, can be thought of as a three-dimensional sphere which grows from a point to a large ball and then shrinks back again. The fifth dimension was color, although I never really understood this (he wasn't very good at explaining it); I think the idea was that, say, red would correspond to a large coordinate in the "color" dimension and blue would correspond to a small coordinate, similar to weather maps showing temperature. So like weather maps allow us to visualize a surface in three-dimensional space with a two-dimensional picture, you can get an extra dimension this way (or even more than one! -- the space of colors is usually thought of as three-dimensional -- although I can't imagine really taking advantage of that).
I suppose a lot of this visualization ability comes with practice, though, and I rarely have any reason to think in more than three dimensions. I can definitely see it coming with practice when I teach calculus students about solids of revolution; at first they just don't seem to get it but eventually they seem to figure it out. The difference there is that I can tell them to draw a picture of the solid in question, and they draw a two-dimensional picture of a three-dimensional thing, which is something we're all used to; drawing a two-dimensional picture of a four-dimensional thing that doesn't throw away all the useful information is a lot more difficult, to the point where perhaps you really have to understand what the thing looks like before you can draw the picture. So they're useful for explaining results to other people, but not so much for coming up with new results.
I've never been good at visualizing things in more than three dimensions (actually, to tell the truth, I have trouble visualizing things in my head in more than two dimensions; to visualize anything successfully in three dimensions I have to use my hands and point at things in the empty space in front of me). But a guy I knew in college claimed to be able to visualize things in five dimensions fairly easily. The fourth dimension was time -- so a four-dimensional sphere, for example, can be thought of as a three-dimensional sphere which grows from a point to a large ball and then shrinks back again. The fifth dimension was color, although I never really understood this (he wasn't very good at explaining it); I think the idea was that, say, red would correspond to a large coordinate in the "color" dimension and blue would correspond to a small coordinate, similar to weather maps showing temperature. So like weather maps allow us to visualize a surface in three-dimensional space with a two-dimensional picture, you can get an extra dimension this way (or even more than one! -- the space of colors is usually thought of as three-dimensional -- although I can't imagine really taking advantage of that).
I suppose a lot of this visualization ability comes with practice, though, and I rarely have any reason to think in more than three dimensions. I can definitely see it coming with practice when I teach calculus students about solids of revolution; at first they just don't seem to get it but eventually they seem to figure it out. The difference there is that I can tell them to draw a picture of the solid in question, and they draw a two-dimensional picture of a three-dimensional thing, which is something we're all used to; drawing a two-dimensional picture of a four-dimensional thing that doesn't throw away all the useful information is a lot more difficult, to the point where perhaps you really have to understand what the thing looks like before you can draw the picture. So they're useful for explaining results to other people, but not so much for coming up with new results.
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