Did you know that did Euler's number, e, is hidden in the Pascal's Triangle?

http://www.brotherstechnology.com/docs/Finding_e_in_Pascal's_Triangle.pdf

#math #maths #pascalstriangle

How to calculate (and display) Pascal's Triangle in TikZ?
Happy 400th BD to Blaise Pascal!
#math #PascalsTriangle
%
%. Draw Pascal's Triangle with binomial coefficients.
%. The triangular numbers are shown in blue.
%
\newcommand \rows {8} % number of rows
%
% Draw the picture
%
\begin{figure}[H]
\centering % center everything
\begin{tikzpicture} [framed,scale=0.90] % frame & scale
\foreach \n in {0,...,\rows} { % iterate over rows
\foreach \k in {0,...,\n} { % iterate over columns
\node at (\k-\n/2,-\n) { % put coefficient here
\ifnum \k = 2 {\color{blue} $\mathbf{\binom{\n}{\k}}$}
\else {\color{black}$\mathbf{\binom{\n}{\k}}$}
\fi}; % end \ifnum
} % end \foreach \k ...
} % end \foreach \n ...
\end{tikzpicture}
\caption{Pascal's Triangle with Binomial Coefficients}
\label{fig:pascals_triangle_binomial_coefficients}
\end{figure}

Today is a great day to remember another one of my favorite mathematicians, Blaise Pascal, who was born #onthisday 400 years ago. Pascal was a French philosopher, mathematician, scientist, inventor, engineer and theologian.

In his short life (he lived only 39 years), he was a pioneer in the fields of game theory and probability theory. In the area of philosophy, he was an early pioneer in existentialism. And as a writer on theology and religion, he was a defender of Christianity.

In 1642 he invented and built the first mechanical calculating machine, the Pascaline, which today is known as Pascal's calculator. He built 20 or so of these beautiful machines (after building 50 prototypes), a few of which survive to this day [1].

And of course, there's the fantastic Pascal's Triangle, a triangular array of the binomial coefficients (and so much more) which Pascal studied and is named in his honor.

#math #pascaline #pascal #pascalstriangle

References
--------------
[1] "Pascal's calculator", https://en.wikipedia.org/wiki/Pascal%27s_calculator

#DidYouKnow: In mathematics, #PascalsTriangle is a triangular array of the binomial coefficients that arises in probability theory, combinatorics, and algebra.

Pascal's triangle determines the coefficients which arise in binomial expansions

https://knowledgezone.co.in/kbits/6429997fc2045832df405e50

A few minor improvements to this picture....

The pdf is here: https://davidmeyer.github.io/qc/tricks.pdf. The LaTeX source is here: https://www.overleaf.com/read/czsqywqhccfw.

As always, questions/comments/corrections/* greatly appreciated.

#math #pascalstriangle #binomialcoefficients

Been drawing some TikZ pictures this morning. It turns out it's pretty easy to draw Pascal's Triangle with TikZ:

%
% Draw Pascal's Triangle with binomial coefficients
%
\begin{figure}[H]
\centering
\begin{tikzpicture} [framed,scale=0.90]
\foreach \n in {0,...,8} {
\foreach \k in {0,...,\n} {
\node at (\k-\n/2,-\n) {
\ifnum \k = 2 {\color{blue} $\binom{\n}{\k}$}
\else {$\binom{\n}{\k}$}
\fi};
}
}
\end{tikzpicture}
\caption{Pascal's Triangle with Binomial Coefficients}
\label{fig:pascals_triangle_binomial_coefficients}
\end{figure}

The pdf is here: https://davidmeyer.github.io/qc/tricks.pdf. The LaTeX source is here: https://www.overleaf.com/read/czsqywqhccfw.

As always, questions/comments/corrections/* greatly appreciated.

#math #pascalstriangle #binomialcoefficients

Some stuff that led to me thinking about a relationship between #PascalsTriangle and #binaryTrees.

It seems to have started with me trying to represent Pascal’s triangle using things like #crochet, #macrame, #braiding and #knitting.

#mathart #mathsart #binaryTree

And another clip of a representation of a #PascalsTriangle #BinaryTree

#mathart

Here is a clip to do with the #PascalsTriangle #BinaryTree shown as individual paths layered together

#mathart

I compiled these ideas into a Twitter moment

#PascalsTriangle

https://twitter.com/i/moments/992234400715862016

Splitting a #woven #PascalsTriangle #BinaryTree into two copies.

#mathart

These ideas eventually worked their way into a #woven #BinaryTree representation of #PascalsTriangle.

#mathart