Visualizing Math

A blog dedicated to mathematics. 

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Spring Forest (5,3): embedded, unembedded, and cowl
12” x 11” x 9”
Knitted wool (Dream in Color Classy, in colors Happy Forest and Spring Tickle)
2009 and 2013
A (p,q) torus knot traverses the meridian cycle of a torus p times and the longitudinal cycle q times. Here are three instantiations of a (5,3) torus knot:
(a, middle) The knot embedded on a torus. A (p,q) torus knot may be drawn on a standard flat torus as a line of slope q/p. The challenge is to design a thickened line with constant slope on a curved surface.
(b, top) The knot projection knitted with a neighborhood of the embedding torus. The knitting proceeds meridianwise, as opposed to the embedded knot, which is knitted longitudinally. Here, one must form the knitting needle into a (5,3) torus knot prior to working rounds.
(c, bottom) The knot projection knitted into a cowl. The result looks like a skinny knotted torus.

Modern Striped Klein
2” x 14” x 7”
Knitted wool (Dream in Color Classy Firescorched in Aqua Jet with Sundown Orchid and Happy Forest)
This Klein bottle was knitted from an intrinsic-twist Mobius band with the boundary self-identified. A Klein bottle can be viewed as the connected sum of two projective planes; here, the stripes highlight the two circles that generate the fundamental groups of the individual projective planes. In some positions, this coloring of the Klein bottle resembles an ouroboros (a snake eating its own tail). The design is more than 10 years old; I recently realized that I had no high-quality example of it (only worn classroom models) and thus created one. Dream in Color veil-dyed yarn was chosen to add a color depth to the seed-stitch texture. Images of this piece graced the cover of the March-April 2013 issue of American Scientist.

Free-Range Mathematician
Sarah Lawrence College / Smith College
Hadley, MA

(via imathematicus)


Did you miss math?

(via imathematicus)


Counting in binary

Instead of counting up to five on each hand, a binary system can be used to count up to 31 on one hand, and up to 1023 on two hands. This is done by using your fingers to represent increasing numbers, multiplying by two each time.

Once the numbers 1, 2, 4, 8 and 16 are assigned to the fingers, as above, different numbers can be represented by raising or tucking in the fingers. A raised finger represents its number being “on”, whereas a lowered finger represents its number being “off”.

For example, raising the thumb (1), the index finger (2) and the ring finger (8) shows a total of 1 + 2 + 8 = 11.

For higher numbers, exactly the same principle is used, by continuing to double the numbers used on the first hand: 32, 64, 128, 256, 512.

Alternatively, by placing your hand above a surface like a table, pressing the fingertip to the surface can be counted as “on”, which is useful both for the less dexterous and for avoiding having the number 4 misinterpreted by somebody else.

(via imathematicus)


The curvature of curves.

  1. sin(x)
  2. exp(x)
  3. Normal distribution (y=exp(-x²/2))
  4. Ellipse
  5. r=5/2+cos(3τθ)
  6. x=(t-1)(t+1), y=t(t-1)(t+1)
  7. Archimedes’ Spiral
  8. Logarithmic spiral

If you want to try your own curve, try on Desmos graphing calculator!

(via spring-of-mathematics)


x^2 + y^2 + z :
x + y^2 + z^2 :
x^2 + y + z^2 :
x^2 - y + z^2 :

x^2 + y^3 + z^3 :
x^2 + y^2 + z :
x + y^2 + z^2 :
x^2 + y + z^2 :
x^2 - y + z^2 :

x^2 + y^3 + z^3 :
x^2 + y^2 + z^3 :
x^2 + y^2 + z :



The animation below shows the first 120 rings of Ulam’s spiral with L varying between 1 and 100:


Below is the corresponding plot of the modified Mertens functionM’(r), for r = 1 to 120:


In mathematical notation, M’ is as follows:


where Rl is the lowest number in the ring with ring index, r, and Rh is the highest number in that ring and μ(k) is the Möbius function.


Below is the corresponding plot of our second modified Mertens function, M”(r), for r = 1 to 120:


In mathematical notation, M” is as follows:



How Mandelbrot’s Fractals Changed The World

by Jack Challoner/BBC News

During the 1980s, people became familiar with fractals through those weird, colorful patterns made by computers. But few realize how the idea of fractals has revolutionized our understanding of the world, and how many fractal-based systems we depend upon.

Unfortunately, there is no definition of fractals that is both simple and accurate. Like so many things in modern science and mathematics, discussions of “fractal geometry” can quickly go over the heads of the non-mathematically-minded. This is a real shame, because there is profound beauty and power in the idea of fractals.

The best way to get a feeling for what fractals are is to consider some examples. Clouds, mountains, coastlines, cauliflowers and ferns are all natural fractals. These shapes have something in common - something intuitive, accessible and aesthetic.

They are all complicated and irregular: the sort of shape that mathematicians used to shy away from in favor of regular ones, like spheres, which they could tame with equations.

Mandelbrot famously wrote: “Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line.”

The chaos and irregularity of the world - Mandelbrot referred to it as “roughness” - is something to be celebrated. It would be a shame if clouds really were spheres, and mountains cones.

Look closely at a fractal, and you will find that the complexity is still present at a smaller scale. A small cloud is strikingly similar to the whole thing. A pine tree is composed of branches that are composed of branches - which in turn are composed of branches.

Read the entire article

Fractal images © Laguna Design / Science Source

Mandelbrodt photo © Emilio Segrè / Science Source

(via mathmajik)


Why can we never learn the interesting side of maths


So here it is. These are the primes of ℤ[√-1], ℤ[√-2], ℤ[(1+√-3)/2], ℤ[√-5], ℤ[√-6], and ℤ[(1+√-7)/2], all in one picture (at the top) and individually (in that order, below.)

The reason these rings are interesting is that each of them consists of solutions to quadratic equations x²+bx+c=0 where b and c are integers and b²-4c is some square times -1, -2, -3, -5, -6, or -7, respectively.

(thx ebering for the idea)


Repeated barycentric subdivision results in a gorgeous fractal-ish pattern.



When people ask me how I can be a math major and still say I’m not good with numbers, I’m like ‘here, let me draw you a picture.’

All so true.

^For those that are considering majoring in math but are deterred by their lack of number skills, there’s definitely more to mathematics than simply numbers. (I’m not dissing stats or number theory or algebra though. Those are cool too!)


Land of mathematics


A roulette traced from rolling an ellipse inside a circle. [thanks] [code]


A roulette traced from rolling an ellipse inside a circle. [thanks] [code]