The Why Phi Pi Slice goes camping.

The leaves of some plants grow on shoots that form 222.5 degrees round the main stem from their predecsessor

Back in 2013 I wrote a post about the relationship between our favourite number, phi (1.618…) and the famous circularity constant Pi (3.141…).

If we divide the circle of 360 degrees by phi, we get 222.5 degrees, leaving 137.5 degrees as the remainder. In that post I noted that:

The area ‘A’ of a sector of a circle is given by the simple formula: A=angle/360*Pi*R²
For a Radius ‘R’ of 1 and angle 137.507764 this is simply 137.5/360*Pi = 1.19998
The area of the whole circle is simply Pi, since R² = 1
The ratio of Pi to 1.19998 is phi²
The ratio of the smaller sector to the larger is Pi-1.19998:1.19998 which is simply phi itself.

While I was idling away some time during the lockdown recently by making a tipi to use later this year, I noticed an interesting further quirk of this relationship between phi and Pi. If you draw a circle on paper and draw a line from the centre to your startpoint, and another line to the point 222.5 degrees round from your startpoint, and then cut out the shape with scissors, you get something looking like this:

Fig1, the phi-Pi slice

If you now join edge a to edge b, you will create a cone with some interesting properties. With a base circle radius of 1, the height of the cone will be the square root of phi, (1.272) and the length from the tip of the cone to the perimeter of the base circle will be phi itself (1.618).

Fig2: Side elevation of a Phi-Pi

Which brings me back to the Tipi I’ve started making. I thought that since the people with the longest experience of getting these abodes to stay up on windy plains are the indigenous north american tribes, it would be a good idea to consult the work of an early anthropologist who bothered to take note of their tipi design.

An Oglala Lakota tipi, 1891

Lewis H. Morgan noted that:

The frame consists of thirteen poles from fifteen to eighteen feet in length, which, after being tied together at the small ends, are raised upright with a twist so as to cross the poles above the fastening. They are then drawn apart at the large ends and adjusted upon the ground in the rim of a circle usually ten feet in diameter.

Note the number of lodge poles is 13, a Fibonacci number. Looking at the geometry of the tipi in the photo, it’s somewhat similar to our phi-Pi slice cone, except the height is phi (1.618) itself rather than root phi (1.272). This produces an elevation profile which is almost an equilateral triangle, with an apex angle of 63.45 degrees and base angles of 58.28 degrees. The sector of a circle required to produce this cone is 189.26 degrees.

Fig3. Real world Tipi vs the phi-pi cone

Two other images I checked had different ratios but also average the same 2:phi for the base diameter to height ratio.

Frances Benjamin JohnstonThe old tipi life among the Sioux1899-1900

Tipis painted by George Catlin in the 1830s

I don’t think the north american tribes used any calculations to decide the proportions of their tipis, but through a process of practical optimisation, arrived at the proportions and dimensions they used taking account of several variables such as family size, straight pole length availability, volume to surface area optimisation (i.e.How to get the most living space out of the least number of buffalo hides), materials economy (especially important if using woven cloth) heating/cooking fire smoke dispersal, and crucially, wind resistance.

Deciding on the optimal cone size and proportion for our own needs as a couple wanting a comfortable place to sleep, dress, cook in bad weather and not wanting to carry too much weight leads to a different set of considerations to compromise amongst. I’ll show you the result soon.