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Aperiodvent, Day 23: The Robin-Lagarias Theorem

Today’s entry is a Theorem of the Day:

The Robin-Lagarias Theorem:

Let $H_n$ denote the n-th harmonic number $\sum_{i=1}^n \frac{1}{i}$ , and let $\sigma(n)$ denote the divisor function $\sum_{d \vert n} d$. Then the Riemann Hypothesis is equivalent to the statement that, for $n \geq 1$, $\sigma(n) \leq H_n + \ln(H_n) e^{H_n}$ .

While this isn’t the traditional Christmas kind of Robin, it is equivalent to the Riemann Hypothesis. For more information, see the full listing at Theorem of the Day: the Robin-Lagarias Theorem.

This is part of the Aperiodical Advent Calendar. We’ll be posting a new surprise for you each morning until Christmas!

 

 

Aperiodvent, Day 22: Gingerbreadman map cookies

Screen Shot 2015-12-13 at 14.30.26The Gingerbreadman Map is a two-dimensional piecewise linear map, defined by:

\begin{align}
x_{n+1} &= 1 – y_n + \lvert x_n \rvert \\
y_{n+1} &= x_n
\end{align}

The region in which the map is chaotic looks like a gingerbread man! In true festive spirit, one blogger has baked some cookies in the shape of the gingerbread man. Check out the blog post describing the cookies, and another one describing the method in more detail.

This is part of the Aperiodical Advent Calendar. We’ll be posting a new surprise for you each morning until Christmas!

Aperiodvent, Day 21: Sierpinski Triangles

siermathgb3Describing itself as ‘the Sierpinski Triangle page to end all Sierpinski Triangle pages‘, this webpage certainly contains an amazing amount of information, diagrams and code to study and explore the well-known equilateral fractal. It goes on forever! (The fractal, not the page – although it does seem like it might never end).

This is part of the Aperiodical Advent Calendar. We’ll be posting a new surprise for you each morning until Christmas!

Aperiodvent, Day 20: a snowflake sequence

 

a161330_caption

Today’s advent calendar window is covered in snowflakes! These snowflakes aren’t your usual sort, however – they’re made up of thousands of toothpicks arranged into E shapes. Hey, nobody mixes metaphors like mathematicians.

The image above shows 1,124 E-shapes arranged rather artfully. 1,124 is the 32nd entry of the sequence A161330, which lists how many shapes are used at each stage of the construction of the snowflake.

The Online Encyclopedia of Integer Sequences has recently made a page showing off many of the prettier pictures contributors have made to illustrate sequences, of which the snowflake above is just one. The idea is that rather than running their own shop selling t-shirts, you can just grab an image off that page and make your own. How in fitting with the giving season!

This is part of the Aperiodical Advent Calendar. We’ll be posting a new surprise for you each morning until Christmas!

Aperiodvent, Day 18: Numbers Aplenty

Screen Shot 2015-12-09 at 19.20.45

Ever wanted to look up a number and find out all kinds of things about it (like, if you’re making a mathematical advent calendar and want interesting facts about the date each day)? No longer do you have to sift through all the tedious non-maths facts you get if you look up a number on Wikipedia: there’s Numbers Aplenty, which allows you to type in any integer up to 15 digits long, and it’ll tell you a long list of the interesting mathematical properties it has. Integeresting!

This is part of the Aperiodical Advent Calendar. We’ll be posting a new surprise for you each morning until Christmas!