Skip to main content

Math: Sierpinski's Triangle is a Variation of Pascal's Triangle

Here's a picture of Pascal's triangle from Wikipedia. It's animated just in case you don't remember how Pascal's triangle is created:

Here's a picture of Sierpinski triangle, also from Wikipedia:

You can see in the top image that to calculate a spot in Pascal's triangle, you just add the two above spots. To get Sierpinski's triangle instead of Pascal's triangle, you just xor the above two spots. Cute trick, eh?

I learned this trick while reading Concepts, Techniques, and Models of Computer Programming, which is a fantastic book, by the way. Here's my Oz code for printing out Pascal's triangle:
% This is a generic version of Pascal's triangle that let's you specify the
% operation instead of just using "+".

declare GenericPascal OpList ShiftLeft ShiftRight
fun {GenericPascal Op N}
if N==1 then [1]
else L in
L={GenericPascal Op N-1}
{OpList Op {ShiftLeft L} {ShiftRight L}}

fun {OpList Op L1 L2}
case L1 of H1|T1 then
case L2 of H2|T2 then
{Op H1 H2}|{OpList Op T1 T2}
else nil end

fun {ShiftLeft L}
case L of H|T then
H|{ShiftLeft T}
else [0] end

fun {ShiftRight L} 0|L end

fun {FastPascal N} {GenericPascal Number.'+' N} end

for I in 1..10 do {Browse {GenericPascal Number.'+' I}} end


Popular posts from this blog

Ubuntu 20.04 on a 2015 15" MacBook Pro

I decided to give Ubuntu 20.04 a try on my 2015 15" MacBook Pro. I didn't actually install it; I just live booted from a USB thumb drive which was enough to try out everything I wanted. In summary, it's not perfect, and issues with my camera would prevent me from switching, but given the right hardware, I think it's a really viable option. The first thing I wanted to try was what would happen if I plugged in a non-HiDPI screen given that my laptop has a HiDPI screen. Without sub-pixel scaling, whatever scale rate I picked for one screen would apply to the other. However, once I turned on sub-pixel scaling, I was able to pick different scale rates for the internal and external displays. That looked ok. I tried plugging in and unplugging multiple times, and it didn't crash. I doubt it'd work with my Thunderbolt display at work, but it worked fine for my HDMI displays at home. I even plugged it into my TV, and it stuck to the 100% scaling I picked for the othe

Drawing Sierpinski's Triangle in Minecraft Using Python

In his keynote at PyCon, Eben Upton, the Executive Director of the Rasberry Pi Foundation, mentioned that not only has Minecraft been ported to the Rasberry Pi, but you can even control it with Python . Since four of my kids are avid Minecraft fans, I figured this might be a good time to teach them to program using Python. So I started yesterday with the goal of programming something cool for Minecraft and then showing it off at the San Francisco Python Meetup in the evening. The first problem that I faced was that I didn't have a Rasberry Pi. You can't hack Minecraft by just installing the Minecraft client. Speaking of which, I didn't have the Minecraft client installed either ;) My kids always play it on their Nexus 7s. I found an open source Minecraft server called Bukkit that "provides the means to extend the popular Minecraft multiplayer server." Then I found a plugin called RaspberryJuice that implements a subset of the Minecraft Pi modding API for B

Creating Windows 10 Boot Media for a Lenovo Thinkpad T410 Using Only a Mac and a Linux Machine

TL;DR: Giovanni and I struggled trying to get Windows 10 installed on the Lenovo Thinkpad T410. We struggled a lot trying to create the installation media because we only had a Mac and a Linux machine to work with. Everytime we tried to boot the USB thumb drive, it just showed us a blinking cursor. At the end, we finally realized that Windows 10 wasn't supported on this laptop :-/ I've heard that it took Thomas Edison 100 tries to figure out the right material to use as a lightbulb filament. Well, I'm no Thomas Edison, but I thought it might be noteworthy to document our attempts at getting it to boot off a USB thumb drive: Download the ISO. Attempt 1: Use Etcher. Etcher says it doesn't work for Windows. Attempt 2: Use Boot Camp Assistant. It doesn't have that feature anymore. Attempt 3: Use Disk Utility on a Mac. Erase a USB thumb drive: Format: ExFAT Scheme: GUID Partition Map Mount the ISO. Copy everything from