Dave Richeson
@divbyzero
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Mathematician. John J. & Ann Curley Chair in Liberal Arts at Dickinson College. Author of Tales of Impossibility and Euler's Gem. Coffee drinker.
Carlisle, PA
Joined August 2008
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Check out my books, "Euler's Gem—The Polyhedron Formula and the Birth of Topology" & "Tales of Impossibility—The 2000-Year Quest to Solve the Mathematical Problems of Antiquity"
@PrincetonUPress
, and my articles
@QuantaMagazine
.
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My friend got these pudding cups. (As in, “the proof is in the pudding.”)
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Cool proof and cool example of using CSS to create mathematical animations on web pages! Here's the page with the math and here's the page describing the CSS .
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Nice puzzle posted by
@icecolbeveridge
on Mastodon (that he said came from Reddit):
Toggle one pixel to make this correct.
No spoilers for those who don't see it!
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A pile of granular material has a maximum slope on its side. This is known as the "angle of repose." I measured the angle of repose of ordinary table salt to be about 33.5°. By pouring salt on raised shapes you can obtain interesting geometric patterns. 1/5
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All of the plenary talks were amazing at Bridges this year!
@mathgrrl
gave the last one, and during the talk she gave everyone a 3D printed copy of this knot, which has the surprising property there does not exist a plane that is tangent to the knot in three places. So it rolls!
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My son, a first-year college students, just called me to say he is going to LaTeX his math notes. ("How do you do piecewise functions?") ♥️♥️
So I had to send him my LaTeX cheat sheet and the Intro to LaTeX video I made
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Topological magic trick: Draw a rectangular border around a piece of paper. Tell the audience member that they will be drawing a simple closed curve on the paper—the more complicated they make it, the better. It can (and should) go in and out of the border rectangle
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I made this fun graphic: I ♥ Math. It is now in the latest issue of the British magazine
@chalkdustmag
(the version appearing there has the British "maths").
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When I was a kid I used to make this sketch: connect (1,0) to (0,10), (2,0) to (0,9), etc. I thought it was cool that these straight lines had a curved boundary. Now I know that this curve is called the *envelope* of the family of lines. What is this curve? 1/7
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LaTeX realhats package—\hat{a} put real hats (dunce cap, Santa hat, top hat, cowboy hat, etc.) on symbols instead of â.
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Why, yes, I did just make this math joke/dad joke for my daughter who is taking calculus right now just to see her eye roll!
"Sky's the limit!"
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A topological magic trick.
(Mathematical explanation in the next tweet.)
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Draw a line from (0,10) to (1,0), from (0,9) to (2,0), and so on. This set of lines has a curved envelope. That curve is a parabola (with equation (x+y−11)²−4xy=0). Here's a still image. Animations in the following tweets.
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I told my son that we are having some of our graduating senior math majors over to our house tomorrow. He said, “When they graduate, do they get degrees… or radians?”
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A nice geometric way of viewing the arithmetic, geometric, and harmonic means of two numbers. It is also easy to see that A>G>H.
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In 1958 Roger Penrose and his father Lionel presented the following interesting "room." The curved walls bounding this room are mirrored, yet a light source placed anywhere in region A will leave region B dark, and vice versa.
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It is really a bummer that math journals require LaTeX. I much prefer writing mathematics in Word.
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Super exciting news about Smith, Meyers,
@cs_kaplan
, Goodman-Strauss's discovery of an "einstein"—a single shape that tiles the plane aperiodically! I decided to create a 3D-printable version of it. You can find it here:
Preprint:
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Wow! I just learned about this online math notebook . I will definitely need to think about how I can use this with my students!
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Theorem. There exist irrational numbers x and y such that x^y is rational.
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My latest print: I designed this polycon—an example of a developable roller. It rolls "straight" albeit in a wobbly zig-zag fashion. Print your own:
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It is possible to raise a transcendental number to an irrational power and obtain a rational number.
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Here's an interesting and famous example that goes back to the 1800s.
Write four nonnegative integers around a circle. Replace each pair of neighboring numbers, a and b, with their absolute difference, |b-a|. We now have four new numbers on the circle. Repeat, repeat,...
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Inspired by the pixel puzzle I posted yesterday, here's one I made.
Toggle TWO pixels (off to on OR on to off) so that this is a true mathematical statement.
Please don't post the solution.
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I had fun making this design, which is my example of an "Euler's spoiler." Here's the math behind it.
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I'm so pleased with this discovery I made a few days ago! I was making a string-art cardioid and wondered how long the string had to be. I knew I could use software to compute the value but thought I'd try calculating the exact length—sure that it would be futile.
It wasn't!
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I found a slip of paper in my desk with this written on it:
61-(8+8+8+8+8)=(8+8+8+8+8)-19 is true and reads the same if rotated 180°.
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In 25 years of teaching calculus, I've never heard the mnemonic BOBO BOTN EATSDC until my daughter uttered it yesterday. It is about limits at infinity of rational functions.
Bigger on Bottom: y=0
Bigger on Top: No [limit]
Exponents Are The Same: Divide Coefficients
Now, I'm
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These salt ridges are good at finding the midline between two shapes. So, pour it over a shape with many punctures and the ridges form a Voronoi diagram. 4/5
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Mathematicians' iPhone passcodes—
Archimedes: 3141
Ramanujan: 1729
Fibonacci: 1123
Euclid: 2357
Brahmagupta: 0000
Peano: 0123
Hippasus: 1414
Pacioli: 1618
Catalan: 1125
Lucas: 2134
Mersenne: 3731
Fermat: 3517
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Sudoku is a problem in graph coloring. Each cell is a vertex, and there are edges from each cell to every cell in its row, column, and 3x3 grid. The objective is to "color" the graph with the numbers/colors 1 to 9 so no two cells joined by an edge are the same color.
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I made another fractal Christmas tree. This one I divided into thirds at each stage rather than halves like the last one.
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I had the idea for this book, Tales of Impossibility, in late fall 2010. Now, almost nine years later, I'm holding it in my hand! Exciting day! The official release date is October 8, but it may arrive in stores before that. Let me know if you see a copy in the wild!
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Pathological function: Volterra's function. This function V is differentiable but V' is not integrable! It seems like this is a counterexample to the Fundamental Theorem of Calculus, but it isn't. In calc textbooks, the FTC requires the function to be continuous, which V' isn't.
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@monamhamdy
I know Amie Wilkinson and Benson Farb (University of Chicago mathematicians) helped come up with the math that appears in the Simpsons' halloween episode! There are some gems in there.
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My 12-year old daughter: What’s the fourth dimension?
Me: [Excitedly tells her about dimension] Why do you ask?
Daughter: I just ran into the corner of the table, and realized that I could see all three dimensions at the same time.
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Here's another roller—an oloid. It is the convex hull of two congruent disks at right angles to each other with their centers one radius apart. Print your own:
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I had a plan for my morning, but when I saw that Smith, Meyers,
@cs_kaplan
, and
@mathbun1
discovered a "spectre"—a CHIRAL aperiodic monotile (one that doesn't require reflections to tile aperiodically)—I knew I had to make one. Here's my 3d-printable file:
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In my calculus lab today I had the students make paper models of a shape they had never seen before. They made traces x=k, y=k, or z=k and then assembled them using either toothpicks or paper bases. Here are two hyperbolic paraboloid models.
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Today in our calculus lab I had the students make paper models of the hyperbolic paraboloid z=y²/2-x²/2. They did so by slicing it at z=const, y=const, or x=const.
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I hadn't done much with with TikZ and pgfplots since this summer. But here's something I made over the weekend—a quick reference to quadric surfaces.
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My daughter: Do you want to hear a dad joke that I think you’ll like...?
I challenged 1 to a fight. He brought 3, 5, 7, and 9 along with him. It was then that I knew the odds were against me.
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Look at this polygon from one side and it looks like a square and from the other side it looks like a diamond. From the side you see it is a non-planar octagon.
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It's common to hear the following facts about Klein bottles:
1) To avoid self-intersections, it must live in at least 4-dimensional space.
2) Like a Möbius band, it's a one-sided surface.
The problem is, a Klein bottle living in 4-dimensional space is NOT one-sided! Why not? 1/13
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The Grammarly web plugin offering to "correct" the phrase "derivative with respect to t."
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My proposed hand sign for a secret math-based organization. The Sierpinski triangle.
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One of my favorite comics—hits my personal sweet spot of math professor and dog owner.
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Assuming f is invertible, knowing the integral of f^{-1} give you the integral for f.
(Saw here: )
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I've spent the last two days teaching myself how to use PGFPlots and TikZ to create visual mathematics inside a LaTeX document. Here are a few things I made while playing around with these tools. So fun!
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Happy 316th birthday to Leohnard Euler! I was just reminded of this Google Doodle from 10 years ago.
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I made this chart for my students to help them clarify in their mind how to interpret the gradient when given a function f or a level set of a function, f=k.
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For instance, pour it on a polygon and it bisects each angle. So when we pour salt on a triangle we obtain a pyramid in which the peak sits above the incenter of the triangle! 2/5
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I went to the hardware store to buy a beaded pull-chain and a solid metal ring so I could perform this math/physics/magic trick.
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The directional derivative of f at each point in ℝ² in the direction of the origin is 0 except at the origin itself where (∂f/∂x)(0,0)=(∂f/∂y)(0,0)=0. Yet f is not a constant function. (And in fact, it is not differentiable at the origin.)
@geogebra
:
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"Is it crazy how saying sentences backward creates backward sentences saying how crazy it is?"
—a word-unit palindrome
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Want a better pencil & paper game than tic-tac-toe? Try Conway/Paterson's Sprouts. Start with n dots. Connect 2 dots or a dot to itself without crossing another curve & so each dot has at most 3 curves coming out. Add a dot in the middle of the curve. Last player with a move wins
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I'll be teaching our intro to proofs class in the fall. This is where our students first learn LaTeX. I spent the last few days making this video for them, "A Quick Introduction to LaTeX." Here's a link to the video: . The topics I cover are shown below.
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What I should be doing: prepping for the start of the semester.
What I am doing: making cardboard conic sections.
Template: from
@MathHappensOrg
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I made this chart to illustrate the three very different notions of size: density, cardinality, and measure. Of the eight options of small (yellow) and large (blue), only two are impossible.
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What do you get if you cut a Möbius band with these crazy 10-bladed scissors (that make five cuts)? Result in the next tweet.
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I'd never seen this painting before—"Mental Arithmetic" (1895) by Nikolay Bogdanov-Belsky. I love it! Alda Carvalho spoke about it and the mental math problem on the board ((10²+11²+12²+13²+14²)/365) at G4G Europe.
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The more I watch this, the more I see <t,cos(t),sin(t)>.
This is near Cromwell, OK. Brent Havins shot this video of a wind turbine that got struck by lightning.
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Here's a great card trick that is easy to perform and will ("probably") amaze your friends. Shuffle a deck of cards. Have your friend draw a card and look at it without showing it to you. In the example below, let's say it is the first card, the 10 of spades. Now, slowly deal
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One way to compute the dimension d of certain shapes:
1) Scale it by a factor of k.
2) This gives you n copies of the original.
3) Then d satisfies n=k^d.
This gives the expected value for a point (0), segment (1), square (2), cube (3). For fractals, the values are not integers.
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Pour the salt on circular shapes or on shapes with circular holes and get conical piles or conical depressions. Where the cones and planes intersect we get conic sections—parabolas, ellipses, and hyperbolas. 3/5
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I'd love to assign credit for this. But I don't know who created it.
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Here's a lovely infinite series that Goldbach told to Euler and Euler wrote about. The proof is clever, but it requires using the divergent harmonic series as if it was convergent. This article has this original proof and a modern rigorous proof.
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Messy hands, messy ruler, and messy paper when I tried using ink. 180 lines: n maps to 4n mod 180.
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In 1971 George Poole discovered the interesting fact that
1!=1,
2!=2,
1!+4!+5!=145, and
4!+0!+5!+8!+5!+=40585,
and that these are the only numbers with this property.
Source:
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Place 2^n spheres inside an n-dimensional cube and one in the middle as shown below. As n grows, so does the radius of the inner sphere. When n=9 the inner sphere is so large it is tangent to the sides of the cube. When n=10 it protrudes out beyond the sides.
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I put a circular piece of paper in the bottom of our flour sifter to get a very clear picture of the reflected cardioid.
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My daughter is taking AP calculus and was talking to me about higher derivatives (and we recently watched All the President's Men). So I had to share this quote.
"In the fall of 1972 President Nixon announced that the rate of increase of inflation was decreasing. This was
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The integers, the rational numbers, the irrational numbers, the real numbers, the algebraic numbers, the transcendental numbers, and the complex numbers. [Diagram not drawn to scale.]
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I just uploaded version 2 of my printable aperiodic tiles. These contain curves on the surface based on the design by
@KangarooPhysics
. You do have to print both versions—the tile and its mirror image (in a ratio of 6 to 1-ish I think).
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Euler proved that π/4=arctan(1/2)+arctan(1/3), which can be used for computing digits of π (using the arctangent series). I wondered if I could prove it without using a trig identity for tangent (which I'd have to look up). Good news: Easy to do using the law of cosines.
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TIL that if we take the average areas of all shadows cast by a convex 3d shape, it is 1/4 the surface area of the shape. The sphere is a trivial case because all shadows are circular disks: S=4(πr^2). This was proved by Cauchy.
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I tried the print-and-cut approach to making some of the aperiodic tiles using a home die cut machine. Now, if only there was a good way to lay out these pieces so they fit together without gaps! 🤣
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Another fascinating and beautifully written article about Euler's mathematics from Bill Dunham in the most recent
@maanow
Monthly. This one is about Euler's attempt to solve the "cubic Basel problem." It contains gems like these.
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On a walk in Vienna today: "The great mathematician and logician Kurt Gödel (1906–1978) lived in this house from 1930 to 1937. Here he discovered his famous incompleteness theorems, the most important mathematical discovery of the twentieth century."
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John von Neumann to the director of the Institute for Advanced Studies in an attempt to get Kurt Gödel out of Austria during WWII: "Gödel is in a class by himself." And "Gödel is absolutely irreplaceable. He is the only mathematician about whom I dare to make this assertion."
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"A man is like a fraction whose numerator is what he is and whose denominator is what he thinks of himself. The larger the denominator the smaller the fraction."—Leo Tolstoy
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The underside of our card table proves the Pythagorean theorem:
c²=4(½ab)+(a-b)²=a²+b².
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point that I know is inside the curve. Following the line from my point to the point A we see it crosses 10 times, so A is inside the curve!
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A little light reading (1056 pages) on this Sunday morning.
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While walking the dog I had the idea for this modification of
@anniek_p
’s
#mathartchallenge
activity. Randomly generate zeros and ones and use them to determine which of the four knot tiles to put in the corresponding cell. Alternate joining the end along the outside. I 1/2
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Neat
@numberphile
video from 2016 in which Cliff Stoll shows, through a progression of glass shapes (!!), that Spivak's "A Hole in a Hole in a Hole" is topologically a 3-holed torus.
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My wife to my son, who's looking at colleges: "How many square acres is that campus?"
Son: "Square acres? Is that its volume?"
Me from the other room: "Actually, that's a four-dimensional measurement."
My wife: "I don't even like you people."
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Looking for something to read during this stay-at-home summer? Interested in mathematics and its history? Try my new book Tales of Impossibility or my 2008 book Euler's Gem, which was recently reissued in the Princeton Science Library series.
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A math friend posted this book cover (published in 2012) on Facebook. I've not seen it before. Lax and Zalcman must have loved coming up with that title!! 🤣
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