Ersby's Triangle

Pascal's Triangle is a famous and fascinating piece of mathematical architecture. In its sequences are many interesting patterns that also appear in the mathematics of probability, fractals, and geometry.

One day, I decided to take a closer look at it. In particular, I wanted to find a formula to describe the numbers along each diagonal.

So far, so good. But I came a bit unstuck when I reached the fourth diagonal. At first, I thought I found something to do with the differences between the numbers...

 Eventually, after several false starts I guessed that since the first two numbers in the sequence were 1 and 4 perhaps the answer was something to do with square numbers. Before long, I came up with something else...

The fifth diagonal was also difficult, but when I went back to working only with square numbers, the answer came quite quickly.

I noticed that 1 squared was being multiplied by the following numbers: 1, 1, 2, 2, 3, 3...

This time the multiples of 1 squared were 1, 2, 4, 6, 9... But I was also interested in how the multiples seemed to "move along" the sum because now 2 squared was being multiplied by 1, 1, 2, 2, 3... I wondered if something was going on so, for clarity's sake, I wrote these multiples on a copy of Pascal's triangle, beside the number that each sum belonged to.

Tiny Manu #89

Sigmund Freud and the analysis of dreams

Tiny Manu #86

It's the cat's meow #108

Based on Schrödinger's cat (a hypothetical scenario used to explain some obscure part of quantum physics) in which a cat is placed in a box with poison and so, as far as the universe is concerned, is both alive and dead until we open the box. It's my attempt at satire and also ties in nicely with tonight's episode of Doctor Who which sees the return of the Weeping Angels, who cannot move if they're being observed in some kind of quantum-style jiggery pokery.

Tiny Manu #77

Happy Valentine's Day

... to all you mathematicians.

Tiny Manu doing sums #2

Tiny Manu doing sums

It seems odd to complain about writer's block when I'm writing as much as I ever did, and especially on a day when I put up three posts, but the fact is that I'm writing fewer comic strips these days. It's been perhaps a month or more since I wrote a six-frame Tiny Manu strip, and my backlog is more or less all used up. However, I'm not abandoning her completely, and here's a little look at something more educational that I've been working on recently. It's still in it's early stages, but I like it.

Tiny Manu #43

Tiny Manu #30

Tomorrow'sWhilred #2

I've long since given up trying to think of anything witty to put on leaving/birthday cards at work. I now just think of something generic like "have a great day". But since everyone else puts that, the challenge then becomes to find the bit of the card furthest away from where anyone else has written the same thing. It can be quite a challenge.

Tiny Manu #26

Tomorrow's Whirled #1

Ah, another new strip. We'll see how long this lasts, although I sometimes come up with science-y things which don't really fit into anything else, so it may be around for a while. Or not.

Tiny Manu's guide to four dimensional space

The Institute for Knowing Things #7

The Institute for Knowing Things #4

The Institute for Knowing Things #3

The Institute for Knowing Things #1

This strip sort of replaced the Gentlemen Scientists after I decided to drop their six frame format. It's similar in that it's very wordy and is sort of based on being clever.

Last of the Gentlemen Scientists #5

The last one - after this I decided to turn my attention to other ideas. Never got round to drawing it properly. Hope you can read my writing.