This is not a food blip - 2

OK, I do know this is silly. In practice you cut food into holdable, mouth-size pieces - and a circle is usually, as ceridwen said yesterday, cut into triangular wedges. But the recipe called for 24 pieces which, with a circle this size, would result in the centre becoming a heap of crumbs. (Yes I know you can just pick them up and eat them but I wanted a prettier solution - and an excuse to play with numbers). I wondered how the whole circle could be cut to get all 24 pieces the same surface area. The cut I've done in the photo is almost what I first envisioned but not quite (because there are 25 pieces in the cut here). I asked myself how large each circle should be and how many pieces each ring should be cut into.

I have a head start here because as well as numbers, I also love spreadsheets so I wrote two simple formulae to find out what would happen to the ring widths if I cut each one into a different number of pieces. 7 or 8 for the inner ring and 15 or 16 for the outer ring would (including the circle in the middle) give me the requisite total of 24. It soon became very clear that 25 pieces would not only be easier to cut but also much neater mathematically: one circle in the middle, eight in the inner ring and 16 (8+8)  in the outer ring. When I put those numbers into the spreadsheet, it showed that the diameter of the innermost circle was equal to the width of both the first and the second rings. Beautiful! Why? There must be a tidy reason for this that I have been deprived of all my life!

This is hard to explain to people who don't love numbers but discovering something that works so neatly that it has to explain other things as well as itself is an endorphin-type pleasure. It was like when I was doodling numbers as a young teenager and the relationships between the differences of squares emerged. (I'll show you that one when we meet.) Maybe it's like playing the killer move towards the end of a game of chess? Or when the missing the right word for a poem comes into your head? Or finding where that errant must-have-come-from-another-box jigsaw piece fits? Or finding the right chord for a piece of music? Or realising that your RNA vaccine works?

I tried to find more online but my searches returned only triangular wedges so yesterday, on a walk in the pouring rain, I thought some more. I realised that this must work for any size circle and any number of rings but I couldn't calculate in my head how many pieces the next ring out would have to be cut into. I worked out the formula I needed to write into the spreadsheet though, and when I got home I tried it. Each ring has eight more pieces than the smaller ring it is next to. For ever. 

Then I realised there's a connection with my teenage doodle (but that's not surprising really because all of maths joins up).

Have a very happy Christmas. You don't need to worry about me, honestly. I shared all this today with my children as we ate the frangipane slices and they loved both. (It's a good thing we have each other.)