What I Haven’t Had Time To Read (Late March)


So here’s a couple things I haven’t had the time to read and think about, but that I want someone to, possibly even me. First, a chain reference:

Paulos’s link in that URL was mistaken and in one of the responses to it he posted a correction. But it’s about this:

And ultimately about what seems a ridiculously impossible condition. Suppose that you have two games, both of which you expect to lose. Or two strategies to play a game, both of which you expect will lose. How do you apply them so that you maximize your chance of winning? Indeed, under the right circumstances, how can you have a better than 50% chance of winning? I have actually read this, but what I haven’t had is the chance to think about it. It may come in handy for pinball league though.

Here, MikesMathPage posts A simplified version of the Banach-Tarski paradox for kids. The Banach-Tarski paradox is one of those things I’m surprised isn’t more common in pop mathematics. It offers this wondrous and absolutely anti-intuitive consequence. Take a sphere the size of a golf ball. Slice it perfectly, using mathematically precise tools that could subdivide atoms, that is, more perfectly than mere matter could ever do. Cut it into pieces and take them apart. Then reassemble the pieces. You have two spheres, and they’re both the size of a planet. You can see why when you get this as a mathematics major the instinct is the say you’ve heard something wrong. There being as many rationals as whole numbers, sure. There being more irratonal numbers than rationals, that’s fine. There being as many points in a one-segment line segment as in an infinitely large ten-dimensional volume of space? Shaky but all right. But this? This? Still, you can kind of imagine that well, maybe there’s some weird thing where you make infinitely many cuts into uncountably infinitely many pieces and then you find out you just need five slices. Four, if you don’t use the point at the very center of the golf ball. Then you get cranky. Anyway the promise of the title, forming a version of this that kids will be comfortable with, is a big one.

This one I’m pretty sure I ended up from by way of Analysis Fact of the day. John D Cook’s Cover time of a graph: cliques, chains, and lollipops is about graphs. I mean graph theory graphs, which look kind of like those circuit-board mass transit diagrams. All dots and lines connecting them. Cook’s question: how long does it take to visit every point in one of these graphs, if you take a random walk? That is, each time you’re at a stop, you take one of the paths randomly? With equal chance of taking any of the paths connected there? There’s some obviously interesting shapes and Cook looks into how you walk over them.

That should do for now. I really need to get caught up on my reading. Please let me know if I’ve made a disastrous mistake with any of this.

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Theorem Thursday: Kuratowski’s Reduction Theorem and Playing With Gas Pipelines


I’m doing that thing again. Sometime for my A To Z posts I’ve used one essay to explain something, but also to introduce ideas and jargon that will come up later. Here, I want to do a theorem in graph theory. But I don’t want to overload one essay. So I’m going to do a theorem that’s interesting and neat in itself. But my real interest in next week’s piece. This is just so recurring readers are ready for it.

The Kuratowski Reduction Theorem.

It starts with a children’s activity book-type puzzle. A lot of real mathematics does. In the traditional form that I have a faint memory of ever actually seeing it’s posed as a problem of hooking up utilities to houses. There are three utilities, usually gas, water, and electricity. There are three houses. Is it possible to connect pipes from each of the three utility starting points to each of the three houses?

Of course. We do it all the time. We do this with many more utilities and many more than three buildings. The underground of Manhattan island is a network of pipes and tunnels and subways and roads and basements and buildings so complicated I doubt humans can understand it all. But the problem isn’t about that. The problem is about connecting these pipes all in the same plane, by drawing lines on a sheet of paper, without ever going into the third dimension. Nor making that little semicircular hop that denotes one line going over the other.

That’s a little harder. By that I mean it’s impossible. You can try and it’s fun to try a while. Draw three dots that are the houses and three dots that are the utilities. Try drawing three lines, one from each utility to each of the houses. Or one leading into each house that comes from each of the utilities. The lines don’t have to be straight. They can have extra jogs, too. Soon you’ll hit on the possibilities of lines that go way out, away from the dots, in the quest to avoid crossing over one another. It doesn’t matter. The attempt’s doomed to failure.

You’ll be sure of this by at latest the twelfth attempt at finding an arrangement. But that leaves open the possibility you weren’t clever enough to find an arrangement. To close that possibility guess what theorem is sitting there ready to answer your question, just like I told you it would be?

This is a problem in graph theory. I’ve talked about graph theory before. It’s the field of mathematics most comfortable to people who like doodling. A graph is a bunch of points, which we call vertices, connected by arcs or lines, which we call edges. For this utilities graph problem, the houses are the vertices. The pipes are the edges. An edge has to start at one vertex and end at a vertex. These may be the same vertex. We’re not judging. A vertex can have one edge connecting it to something else, or two edges, or three edges. It can have no edges. It can have any number of edges. We’re even allowed to have two or more edges connecting a vertex to the same vertex. My experience is we think of that last, forgetting that it is a possibility, but it’s there.

This is a “nonplanar” graph. This means you can’t draw it in a plane, like a sheet of paper, without having at least two edges crossing each other. We draw this on paper by making one of the lines wiggle in a little half-circle to show it’s going over, or to fade out and back in again to show it’s going under. There are planar graphs. Try the same problem with two houses and two utilities, for example. Or three houses and two utilities. Or three houses and three utilities, but one of the houses doesn’t get one of the utilities. Your choice which. It can be a little surprise to the homeowners.

This utilities graph is an example of a “bipartite” graph. The “bi” maybe suggests where things are going. You can always divide the vertices in a graph into two groups for the same reason you can always divide a pile of change into two piles. As long as you have at least two vertices or pieces of change. But a graph is bipartite if, once you’ve divided the vertices up, each edge has one vertex in the first set and the other vertex in the second. For the utilities graph these sets are easy to find. Each edge, each pipe, connects one utility to one house. There’s our division: vertices representing houses and vertices representing utilities.

This graph turns up a lot. Graph theorists have a shorthand way of writing it. It’s written as K3,3. This means it’s a bipartite graph. It has three vertices in the first set. There’s three vertices in the second set. There’s an edge connecting everything in the first set to everything in the second. Go ahead now and guess what K2, 2 is. Or K3,5. The K — I’ve never heard what the K stands for, although while writing this essay I started to wonder if it’s for “Kuratowski”. That seems too organized, somehow.

Not every graph is bipartite. You might say “of course; why else would we have a name `bipartite’ if there weren’t such a thing as `non-bipartite’?” Well, we have the name “graph” for everything that’s a graph in graph theory. But there are non-bipartite graphs. They just don’t look like the utility-graph problem. Imagine three vertices, each of them connected to the other two. If you aren’t imagining a triangle you’re overthinking this. But this is a perfectly good non-bipartite graph. There’s no way to split the vertices into two sets with every edge connecting something in one set to something in the other. No, that isn’t inevitable once you have an odd number of vertices. Look above at the utilities problem where there’s three houses and two utilities. That’s nice and bipartite.

Non-bipartite graphs can be planar. The one with three vertices, each connected to each other, is. The one with four vertices, each vertex connected to each other, is also planar. But if you have five vertices, each connected to each other — well, that’s a lovely star-in-pentagon shape. It’s also not planar. There’s no connecting each vertex to each other one without some line crossing another or jumping out of the plane.

This shape, five vertices each connected to one another, shows up a lot too. And it has a shorthand notation. It’s K5. That is, it’s five points, all connected to each other. This makes it a “complete” graph: every set of two vertices has an edge connecting them. If you’ve leapt to the supposition that K3 is that circle and K4 is that square with diagonals drawn in you’re right. K6 is six vertices, each one connected to the five other vertices.

It may seem intolerably confusing that we have two kinds of graphs and describe them both with K and a subscript. But they’re completely different. The bipartite graphs have a subscript that’s two numbers separated by a comma: p, q. The p is the number of vertices in one of the subsets. The q is the number of vertices in the other subset. There’s an edge connecting every point in p to every point in q, and vice-versa. The points in the p subset aren’t connected to one another, though. And the points in the q subset aren’t connected to one another. That they don’t mean this isn’t a complete graph.

The others, K with a single number r in the subscript, are complete graphs, ones that aren’t bipartite. They have r vertices, and each vertex is connected to the (r – 1) other vertices. So there’s (1/2) times r times (r – 1) edges all told.

Not every graph is either Kp, q or Kr. There’s a lot of kinds of graphs out there. Some are planar, some are not. But here’s an amazing thing, and it’s Kuratowski’s Reduction Theorem. If a graph is not planar, then it has to have, somewhere inside it, K3, 3 or K5 or both. Maybe several of them.

A graph that’s hidden within another is called a “subgraph”. This follows the same etymological reasoning that gives us “subsets” and “subgroups” and many other mathematics words beginning with “sub”. And these subgraphs turn up whenever you have a nonplanar graph. A subgraph uses some set of the vertices and edges of the original graph; it doesn’t need all of them. A nonplanar graph has a subgraph that’s K3, 3 or K5 or both.

Sometimes it’s easy to find one of these. K4, 4 obviously has K3, 3 inside it. Pick three of the four vertices on one side and three of the four vertices on the other, and look at the edges connecting them up. There’s your K3, 3. Or on the other side, K6 obviously has K5 inside it. Pick any five of the vertices inside K6 and the edges connecting those vertices. There’s your K5.

Sometimes it’s hard to find one of these. We can make a graph look more complicated without changing whether it’s planar or not. Take your K3, 3 again. Go to each edge and draw another dot, another vertex, inside it. Well, now it’s a graph that’s got twelve vertices in it. It’s not obvious whether this is bipartite still. (Play with it a while.) But it hasn’t become planar, not because of this. It won’t be.

This is because we can make graphs more complicated, or more simple, without changing whether they’re planar. The process is a lot like what we did last week with the five-color map theorem, making a map simpler until it was easy enough to color. Suppose there’s a little section of the graph that’s a vertex connected by one edge to a middle vertex connected by one edge to a third vertex. Do we actually need that middle vertex for anything? Besides padding our vertex count? Nah. We can drop that whole edge-middle vertex-edge sequence and replace it all with a single edge. And there’s other rules that let us turn a vertex-edge-vertex set into a single vertex. That sort of thing. It won’t change a planar graph to a nonplanar one or vice-versa.

So it can be hard to find the K3, 3 or the K5 hiding inside a nonplanar graph. A big enough graph can have so much going on it’s hard to find the pattern. But they’ll be there, in some group of five or six vertices with the right paths between them.

It would make a good activity puzzle, if you could explain what to look for to kids.

Reading the Comics, April 24, 2016: Mental Mathematics and Calendars Edition


Warning! I do some showing off in this installment of the Reading the Comics series. Please forgive me. I was feeling a little giddy.

Scott Hilburn’s The Argyle Sweater I had just mentioned to a friend never seems to show up in these columns anymore. And Hilburn would so reliably do strips about anthropomorphized numerals. He returns on the 20th, after a hiatus of some length I haven’t actually checked here, with a name-drop of Einstein instead. I grinned, although a good part of what amused me was the look of the guy in the lower right of the panel. Funny pictures carry a comic strip far. Formulating the theory of relativity is a tricky request. The special theory … well, to do it properly takes some sophisticated work. But it doesn’t take much beyond the Pythagorean Theorem to realize that “how long” a thing is, or a time span is, is different for different observers. That’s the most important insight, I would say, and that is easily available. General relativity, which looks at accelerations and gravity, that’s another thing. I’d be interested in a popular treatment that explained enough mathematics people could make usable estimates but that could still make sense to a lay audience. Probably it’s not possible to do this. Too bad.

Mark Tatulli’s Heart of the City just uses arithmetic because it’s a nice compact problem to give a student. It did strike me that 117 times 45 is something one could amaze people with by doing in one’s head, though. Here’s why. 117 times 100 would be easy. Multiplying by hundreds always is. 117 times 50 would be not almost as easy: that’s multiplying by 100 and dividing by two. 117 times 45 … well, that’s 117 times 50 minus 117 times 5. And if you know 117 times 50, then you know 117 times 5: it’s one-tenth that. And one-tenth of a thing is easy to find.

Therefore: 117 times 100 is 11,700. Divide that by two and that’s kind of an ugly-looking number, isn’t it? But all’s not lost. Let me use another bit of falsework: 11,700 is 12,000 minus 300. Half that is 6,000 minus 150. Therefore, half of 11,700 is 5,850. So 117 times 50 is 5,850. One-tenth of that is 585. Therefore, 117 times 45 is 5,850 minus 585. And that will be … 5,275. Ta-da!

Well, no, it isn’t. It’s 5,265. I messed up the carrying. I still think that’s doing well for multiplying ugly numbers like that without writing it down. It just won’t impress people who want the actual you know correct answer.

Mark Anderson’s Andertoons wouldn’t let me down by vanishing for a while. The 21st is not explicitly a strip about extrapolating graphs. I’ll take it as such, though. Once again the art amuses me. I like the crash-up of charted bars. Yes, I saw the Schrödinger’s Cat thing two days later.

Jef Mallett’s Frazz for the 23rd I drag into a mathematics blog because of the long historical links between calendars and mathematics. But Caulfield does talk about something that’s baffled everyone. There’s seven days to the week. There’s seven classically known heavenly bodies in the solar system, besides the Earth. Naming a day for each seems obvious now that we’ve committed to it. But why aren’t the bodies honored in order?

Geocentrism seems like, at first, a plausible reason. The ancients wouldn’t order the sky Sun-Mercury-Venus-Moon-Mars-Jupiter-Saturn. But that doesn’t help. Geocentric models of the solar system (always, so far as I’m aware) put the Moon closest, then Mercury, then Venus, the Sun, Mars, Jupiter, and Saturn.

The answer that, at least, gets repeated in histories of the calendar (for example, here, David Ewing Duncan’s The Calendar: The 5000-Year Struggle To Align The Clock And The Heavens — And What Happened To The Missing Ten Days, which was the first book I had on hand) amounts to a modular arithmetic thing. The Babylonians, if Duncan is right, named a planet-god for each hour of the day. (We treat the Moon and Sun as planets for this discussion.) The planet-gods took their hourly turn in order. If the first hour of the day is Saturn’s to rule, the next is Jupiter’s, then Mars’s, the Sun’s, Venus’s, Mercury’s, and the Moon’s. Then back to Saturn and the system keeps going like that.

So if the first hour of the day is Saturn’s, then who has the first hour of the next day? … the Sun does. If the Sun has the first hour of the day, then who has the first hour of the day after that? … the Moon. And from here you know the pattern. At least you do if you understand that English derives most of its day names from the Norse gods, matched as best they can with those of the Roman State Religion. So, Tiw matches with Mars; Woden with Mercury; Thor with Jupiter; Freya with Venus. The apparently scrambled order of days, relative to the positions of the planets, amounts to what you get if you keep adding 24 to a number by modulo 7 arithmetic.

That is, at least, the generally agreed-upon explanation. I am not aware of what actual researchers of Babylonian culture believe. Duncan, I must admit, takes a hit in his credibility by saying on the page after this that “recently chronobiologists have discovered that the seven-day cycle … may also have biological precedents”. I’m sorry but I just don’t believe him, or whoever he got that from.

Kevin Fagan’s Drabble for the 24th amuses me by illustrating the common phenomenon. We have all taken out the calculator (or computer) to do some calculation that really doesn’t need it. I understand and am sympathetic. It’s so obviously useful to let the calculator work out 117 times 45 and get it right instantly. It’s easy to forget sometimes it’s faster to not bother with the calculator. We are all of us a little ridiculous.

A Leap Day 2016 Mathematics A To Z: X-Intercept


Oh, x- and y-, why are you so poor in mathematics terms? I brave my way.

X-Intercept.

I did not get much out of my eighth-grade, pre-algebra, class. I didn’t connect with the teacher at all. There were a few little bits to get through my disinterest. One came in graphing. Not graph theory, of course, but the graphing we do in middle school and high school. That’s where we find points on the plane with coordinates that make some expression true. Two major terms kept coming up in drawing curves of lines. They’re the x-intercept and the y-intercept. They had this lovely, faintly technical, faintly science-y sound. I think the teacher emphasized a few times they were “intercepts”, not “intersects”. But it’s hard to explain to an eighth-grader why this is an important difference to make. I’m not sure I could explain it to myself.

An x-intercept is a point where the plot of a curve and the x-axis meet. So we’re assuming this is a Cartesian coordinate system, the kind marked off with a pair of lines meeting at right angles. It’s usually two-dimensional, sometimes three-dimensional. I don’t know anyone who’s worried about the x-intercept for a four-dimensional space. Even higher dimensions are right out. The thing that confused me the most, when learning this, is a small one. The x-axis is points that have a y-coordinate of zero. Not an x-coordinate of zero. So in a two-dimensional space it makes sense to describe the x-intercept as a single value. That’ll be the x-coordinate, and the point with the x-coordinate of that and the y-coordinate of zero is the intercept.

If you have an expression and you want to find an x-intercept, you need to find values of x which make the expression equal to zero. We get the idea from studying lines. There are a couple of typical representations of lines. They almost always use x for the horizontal coordinate, and y for the vertical coordinate. The names are only different if the author is making a point about the arbitrariness of variable names. Sigh at such an author and move on. An x-intercept has a y-coordinate of zero, so, set any appearance of ‘y’ in the expression equal to zero and find out what value or values of x make this true. If the expression is an equation for a line there’ll be just the one point, unless the line is horizontal. (If the line is horizontal, then either every point on the x-axis is an intercept, or else none of them are. The line is either “y equals zero”, or it is “y equals something other than zero”. )

There’s also a y-intercept. It is exactly what you’d imagine once you know that. It’s usually easier to find what the y-intercept is. The equation describing a curve is typically written in the form “y = f(x)”. That is, y is by itself on one side, and some complicated expression involving x’s is on the other. Working out what y is for a given x is straightforward. Working out what x is for a given y is … not hard, for a line. For more complicated shapes it can be difficult. There might not be a unique answer. That’s all right. There may be several x-intercepts.

There are a couple names for the x-intercepts. The one that turns up most often away from the pre-algebra and high school algebra study of lines is a “zero”. It’s one of those bits in which mathematicians seem to be trying to make it hard for students. A “zero” of the function f(x) is generally not what you get when you evaluate it for x equalling zero. Sorry about that. It’s the values of x for which f(x) equals zero. We also call them “roots”.

OK, but who cares?

Well, if you want to understand the shape of a curve, the way a function looks, it helps to plot it. Today, yeah, pull up Mathematica or Matlab or Octave or some other program and you get your plot. Fair enough. If you don’t have a computer that can plot like that, the way I did in middle school, you have to do it by hand. And then the intercepts are clues to how to sketch the function. They are, relatively, easy points which you can find, and which you know must be on the curve. We may form a very rough sketch of the curve. But that rough picture may be better than having nothing.

And we can learn about the behavior of functions even without plotting, or sketching a plot. Intercepts of expressions, or of parts of expressions, are points where the value might change from positive to negative. If the denominator of a part of the expression has an x-intercept, this could be a point where the function’s value is undefined. It may be a discontinuity in the function. The function’s values might jump wildly between one side and another. These are often the important things about understanding functions. Where are they positive? Where are they negative? Where are they continuous? Where are they not?

These are things we often want to know about functions. And we learn many of them by looking for the intercepts, x- and y-.

A Summer 2015 Mathematics A To Z: vertex (graph theory)


Vertex.

I mentioned graph theory several weeks back, when this Mathematics A To Z project was barely begun. It’s a fun field. It’s a great one for doodlers, and it’s one that has surprising links to other problems.

Graph theory divides the conceptual universe into “things that could be connected” and “ways they are connected”. The “things that could be connected” we call vertices. The “ways they are connected” are the edges. Vertices might have an obvious physical interpretation. They might, represent the corners of a cube or a pyramid or some other common shape. That, I imagine, is why these things were ever called vertices. A diagram of a graph can look a lot like a drawing of a solid object. It doesn’t have to, though. Many graphs will have vertices and edges connected in ways that no solid object could have. They will usually be ones that you could build in wireframe. Use gumdrops for the vertices and strands of wire or plastic or pencils for the edges.

Vertices might stand in for the houses that need to be connected to sources of water and electricity and Internet. They might be the way we represent devices connected on the Internet. They might represent all the area within a state’s boundaries. The Köningsburg bridge problem, held up as the ancestor of graph theory, has its vertices represent the islands and river banks one gets to by bridges. Vertices are, as I say, the things that might be connected.

“Things that might be connected” is a broader category than you might imagine. For example, an important practical use of mathematics is making error-detecting and error-correcting codes. This is how you might send a message that gets garbled — in sending, in transmitting, or in reception — and still understand what was meant. You can model error-detecting or correcting codes as a graph. In this case every possible message is a vertex. Edges connect together the messages that could plausibly be misinterpreted as one another. How many edges you draw — how much misunderstanding you allow for — depends on how many errors you want to be able to detect, or to correct.

When we draw this on paper or a chalkboard or the like we usually draw it as a + or an x or maybe a *. How much we draw depends on how afraid we are of losing sight of it as we keep working. In publication it’s often drawn as a simple dot. This is because printers are able to draw dots that don’t get muddied up by edges being drawn in or eraser marks removing edges.

A Summer 2015 Mathematics A To Z: jump (discontinuity)


Jump discontinuity.

Analysis is one of the major subjects in mathematics. That’s the study of functions. These usually have numbers as the domain and the range. The domain and range might be the real numbers, or complex numbers, or they might be sets of real or complex numbers. But they’re all numbers. If you asked for an example of one of these functions you’d get something that looked more or less like a function out of high school.

Continuity is one of the things mathematicians look for in functions. To a mathematician continuity means almost what you’d imagine from the everyday definition of the term. You could draw a sketch of a continuous function without having to lift your pen off the paper. (Typically. If you want to, you can define functions that meet the proper mathematical definition of “continuous” but that you really can’t draw. Mathematicians use these functions to keep one another humble.)

Continuous functions tend to be nice ones to work with. Continuity usually makes it easier to prove a function has whatever other properties you’d like. Mathematicians will even talk about continuous functions as being nice and well-behaved and even normal, as though the functions being easier to work with bestowed on them some moral virtue. However, not every function is continuous. Properly speaking, most functions aren’t continuous. This is the same way that most numbers aren’t whole numbers.

There are different ways that a function can be discontinuous. One of the easiest to understand and to work with is called a jump discontinuity. If you draw a plot representing a function with a jump discontinuity, it looks rather like the plot of a nice, well-behaved, continuous function except that at the discontinuity it jumps. From one side of the discontinuity to the other the function suddenly hops upward, or drops downward.

If a function only has jump discontinuities we aren’t badly off. We can write a function with jump discontinuities as the sum of a continuous function and a function made up only of jumps. The continuous function will be easy to work with, since it’s continuous. The function made of jumps isn’t continuous, by definition, but it’s going to be “flat” — it’ll have the same value in-between any two jumps. That’s usually easy to work with, and while the details of these jump functions will be different they’ll all look about the same. They’ll have different heights and jump up or down at different points, but if you know how to understand a function that jumps from being equal to 0 to being equal to 1 when the input goes from just below to just above 2, then you know how to understand a function that jumps from being equal to 0 to being equal to 3 when the input goes from just below 2.5 to just above 2.5.

This won’t let us work with every function. Most functions are going to be discontinuous in ways that we can’t resolve with jump functions. But a lot of the functions we’re naturally interested in, because they model interesting problems, can be. And so we can divide tricky functions into sets of functions that are easier to deal with.

A Summer 2015 Mathematics A To Z: graph


Graph. (As in Graph Theory)

When I started this A to Z I figured it would be a nice string of quick, two-to-four paragraph compositions. So far each one has been a monster post instead. I’m hoping to get to some easier ones. For today I mean to talk about a graph, as in graph theory. That’s not the kind of graph that’s a plot of some function or a pie chart or a labelled map or something like that.

This kind of graph we do study as pictures, though. Specifically, they’re pictures with two essential pieces: a bunch of points or dots and a bunch of curves connecting dots. The dots we call vertices. The curves we call edges. My mental model tends to be of a bunch of points in a pegboard connected by wire or string. That might not work for you, but the idea of connecting things is what graphs, and graph theory, are good for studying.

Continue reading “A Summer 2015 Mathematics A To Z: graph”

Duality, fundamental and profound, but here’s a starter for you.


This week in the mathematics A-To-Z I’ve been writing I mentioned duals. I asserted there were duals all over mathematics, but gave only one example, that of turning solid shapes into other solid shapes. It happens that two weeks ago HowardAt58’s “Saving School Math” blog ran a post, with pictures, about creating duals to lines, and to points on the plane. If you’re careful to set out the rules you start with, you can match any straight line to a point in the plane. And you can match any point in the plane to a straight line. And so … well, read, and see some ways to look at lines and points and other shapes which may be mind-expanding.

Saving school math

Duality, how things are connected in unexpected ways. The simplest case is that of the five regular Platonic solids, the tetrahedron, the cube, the octahedron, the dodecahedron and the icosahedron. They all look rather different, BUT…..

take any one of them and find the mid point of each of the faces, join these points up, and you get one of the five regular Platonic solids. Do it to this new one and you get back to the original one. Calling the operation “Doit” we get

tetrahedron –Doit–> tetrahedron –Doit–> tetrahedron
cube –Doit–> octahedron –Doit–> cube
dodecahedron –Doit–> icosahedron –Doit–> dodecahedron

The sizes may change, but we are only interested in the shapes.

This is called a Duality relationship, in which the tetrahedron is the dual of itself, the cube and octahedron are duals of each other, and the dodecahedron and icosahedron are also duals of each other.

Now we will look at lines…

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Earth Day


It’s a lovely day, so I felt like sharing these illustrations from the Life Through A Mathematicians Eyes blog. It’s simply superimposing graphs of equations over scenes of natural beauty, but that’s attractive enough.

When I say “graphs of equations”, I mean that we’re setting a coordinate system — here a Cartesian or rectangular one, one based on x- and y- and z- distances from some origin point — over space, and then drawing in white lines the sets of x- and y- and z-coordinates which make some equation true. That’s what we normally mean by saying “the graph of an equation”; it’s a drawing that shows when a relationship is true and when it is not.

Life Through A Mathematician's Eyes

I believe most of you know what Earth Day is celebrating ^_^ I think this is a great day and there are a lot of activities that could be done everywhere to celebrate it. You might think that there is not much I can say about Earth and mathematics, but if you are a math – lover like you already know a lot of mathematical shapes, patterns and constant that can be found in nature. If you want to see more about this check the photos from my album Math&Nature .

But today I want to talk a little about the idea of an American student, mathematician and photographer, Nikki Graziano. I believe the project is old, but still very interesting and perfect for today. She took photos of different natural forms and then found proper equations to explain those forms created by nature. Here are some of the images:

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Reading the Comics, February 28, 2015: Calendar Reform Edition


It’s the last day of the shortest month of the year, a day that always makes me think about whether the calendar could be different. I was bit by the calendar-reform bug as a child and I’ve mostly recovered from the infection, but some things can make it flare up again and I’ve never stopped being fascinated by the problem of keeping track of days, which you’d think would not be so difficult.

That’s why I’m leading this review of comics with Jef Mallet’s Frazz (February 27) even if it’s not transparently a mathematics topic. The biggest problem with calendar reform is there really aren’t fully satisfactory ways to do it. If you want every month to be as equal as possible, yeah, 13 months of 28 days each, plus one day (in leap years, two days) that doesn’t belong to any month or week is probably the least obnoxious, if you don’t mind 13 months to the year meaning there’s no good way to make a year-at-a-glance calendar tolerably symmetric. If you don’t want the unlucky, prime number of 13 months, you can go with four blocks of months with 31-30-30 days and toss in a leap day that’s again, not in any month or week. But people don’t seem perfectly comfortable with days that belong to no month — suggest it to folks, see how they get weirded out — and a month that doesn’t belong to any week is right out. Ask them. Changing the default map projection in schools is an easier task to complete.

There are several problems with the calendar, starting with the year being more nearly 365 days than a nice, round, supremely divisible 360. Also a factor is that the calendar tries to hack together the moon-based months with the sun-based year, and those don’t fit together on any cycle that’s convenient to human use. Add to that the need for Easter to be close to the vernal equinox without being right at Passover and you have a muddle of requirements, and the best we can hope for is that the system doesn’t get too bad.

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Before Drawing a Graph


I want to talk about drawing graphs, specifically, drawing curves on graphs. We know roughly what’s meant by that: it’s about wiggly shapes with a faint rectangular grid, usually in grey or maybe drawn in dotted lines, behind them. Sometimes the wiggly shapes will be in bright colors, to clarify a complicated figure or to justify printing the textbook in color. Those graphs.

I clarify because there is a type of math called graph theory in which, yes, you might draw graphs, but there what’s meant by a graph is just any sort of group of points, called vertices, connected by lines or curves. It makes great sense as a name, but it’s not what what someone who talks about drawing a graph means, up until graph theory gets into consideration. Those graphs are fun, particularly because they’re insensitive to exactly where the vertices are, so you get to exercise some artistic talent instead of figuring out whatever you were trying to prove in the problem.

The ordinary kind of graphs offer some wonderful advantages. The obvious one is that they’re pictures. People can very often understand a picture of something much faster than they can understand other sorts of descriptions. This probably doesn’t need any demonstration; if it does, try looking at a map of the boundaries of South Carolina versus reading a description of its boundaries. Some problems are much easier to work out if we can approach it as a geometric problem. (And I admit feeling a particular delight when I can prove a problem geometrically; it feels cleverer.)

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