Access and a Fibonacci wedding cake
- Date: 22 June 2023
- Time: 6:30 to 7:30pm (UK Time)
Summary
Jonathan Fine This is an informal personal talk about the influence of accessibility on a math research problem, and vice versa. The problem is to define something like Pascal’s triangle, but with the rows summing to the Fibonacci numbers. Such I call a Fibonacci wedding cake, as it is an elaborate tower made out of multiple tiers.
Pascal’s triangle
Here’s something that surprised me. Pascal’s triangle can be presented in several ways. For a sighted person the usual way (using Markdown) is:
| 1 | ||||||||
| 1 | 1 | |||||||
| 1 | 2 | 1 | ||||||
| 1 | 3 | 3 | 1 | |||||
| 1 | 4 | 6 | 3 | 1 |
The surprise was the thought that for a blind person (again in Markdown) this left-aligned form might be much better.
| 1 | ||||
| 1 | 1 | |||
| 1 | 2 | 1 | ||
| 1 | 3 | 3 | 1 | |
| 1 | 4 | 6 | 4 | 1 |
I’d very much welcome any feedback on access to number triangles. Although the Fibonacci wedding cake is connected to some advanced topics, no special knowledge of pure math is required for this talk. Just an interest in learning about and discovering patterns.
Fibonacci numbers
The Fibonacci numbers are 0, 1, 2, 3, 4, 5, 8, 13 and so on. The next Fibonacci number is the sum of the two previous. For example 5 + 8 = 13 and 8 + 13 = 21.
We write each Fibonacci number as a sum, analogous in some ways to Pascal’s triangle. Each row is constructed from the two previous rows. (Pascal’s triangle uses only the previous row.)
The problem is to group together the numbers in the previous two rows, to get the next row. For example 2 = 1 + 1 and 3 = 1 + 1 + 1. Now write 2 + 3 = 5 = 1 + (1 + 1 + 1) + 1 = 1 + 3 + 1.
Fibonacci conventions
- Each tier has 3 rows, all the same length.
- The tier lengths are 1, 3, 5, 7, 9, 11 and so on.
- The top row of each tier has a gap in the middle, denoted by
.. - We use an
xto denote the end of a row
Fibonacci numbers 0, 1, 1, 2, 3, 5, 8, 13, 21, 34
| 0 = | . | x | ||||
| 1 = | 1 | x | ||||
| 1 = | 1 | x | ||||
| 2 = | 1 | . | 1 | x | ||
| 3 = | 1 | 1 | 1 | x | ||
| 5 = | 1 | 3 | 1 | x | ||
| 8 = | 1 | 3 | . | 3 | 1 | x |
| 13 = | 1 | 3 | 5 | 3 | 1 | x |
| 21 = | 1 | 4 | 11 | 4 | 1 | x |
Example: 13 + 21 = 34
This calculation gives the top row of the next tier.
| 13 = | 1 | 3 | 5 | 3 | 1 | x | ||
| 21 = | 1 | 4 | 11 | 4 | 1 | x | ||
| 34 = | 1 | 5 | 11 | . | 11 | 5 | 1 | x |
Exercise: 21 + 34 = 55
| 21 = | 1 | 4 | 11 | 4 | 1 | x | ||
| 34 = | 1 | 5 | 11 | . | 11 | 5 | 1 | x |
| 55 = | 1 | 6 | ? | ? | ? | ? | ? | x |
URLs
Real world wedding cakes
- Photo of Fibonacci Wedding Cake (Aperiodical math blog)
- Wedding cakes (1947) for Princess Elizabeth and Prince Philip (wikipedia)
- Bakers prepare royal wedding cake (1947: British Pathe video)