The Why Phi Pi Slice goes camping.

Posted: May 12, 2020 by tallbloke in design, Phi
Tags: , ,
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.

Comments
  1. hunterson7 says:

    A lot of food for thought. Thank you.

  2. oldbrew says:

    A cone is like a pyramid…
    https://www.mathsisfun.com/geometry/pyramid-vs-cone.html

    The volume of a pyramid (also any cone) is {see link}, where b is the area of the base and h the height from the base to the apex.
    https://en.wikipedia.org/wiki/Pyramid_(geometry)#Volume

  3. […] über The Why Phi Pi Slice goes camping. — Tallbloke’s Talkshop […]

  4. tallbloke says:

    A cone is a pyramid with an infinite number of sides. That makes it the most efficient of the pyramids in terms of the surface area to volume ratio (SAR). The most efficient cone has a height three times the base radius. Such a cone holding 1 litre would have a surface area of 610cm^2

    Comparing the surface areas of cones, spheres and cylinders is interesting, check the equations here: https://www.mathsisfun.com/geometry/cone-sphere-cylinder.html

    Another curiosity is that (1+root5)/6 = ‭0.53934466 which is also the volume of a cone of base radius 1 and a height of phi!

  5. oldbrew says:

    ‭0.53934466 = 1.618034 / 3

    The interesting bit is the use of root5.

    Different surface area formula here?

    https://www.onlinemathlearning.com/surface-area-cone.html

  6. tallbloke says:

    I thought that 0.593 looked familiar but didn’t spot it was a third of phi. The (1+root5)/6 isn’t the whole formula for cone surface area, it’s what’s left after the canceling out done across the comparison equations for sphere and cylinder. The 1+root5 x Pi.r^2 as the total area (with the base circle area) is a special case for a cone of height 2x base circle radius. However that in itself is interesting, because it’s another obvious link between phi and pi.

    In this special case, we’re back to Paul Vaughan’s 2 x 1 Conway triangle with root5 as the hypotenuse.

  7. oldbrew says:

    Right, your tipi is an upright right-angled triangle sweeping round on its circular base with height = diameter = 2*radius.

    The Catlin tipi looks equilateral, which would make the height root3 (1.732).

  8. tallbloke says:

    I can’t remember how I constructed root3 on this diagram Paul V originated, but there are the 2 x 1 triangles, root 5 and the phi relationship. Naturally, the circumference of the yellow circle is Pi.

    EDIT: Ah yes, Root3 is constructed by drawing an R=1 circle and then drawing a tangential line from 2R away from the centre to a point on the circumference. I measure Catlin at 1.77 but who knows what artistic license he used.

  9. tallbloke says:

    Here’s a simpler geometrical construction I’ve done capturing some of what we’ve discovered about teepees and cones, root5 and phi.

  10. tallbloke says:

    In the special case that the base diameter is equal to the slope length of the cone (an equilateral triangle as seen from the side), the lateral area forms a semicircle and total area is given by Pi*r^2+Pi*3r. For a base diameter of 2 the area is simply 3*Pi

  11. oldbrew says:

    This time a ‘Tepe’…

    Geometry guided construction of earliest known temple, built 6,000 years before Stonehenge
    May 12, 2020 , Tel Aviv University

    ‘Researchers at Tel Aviv University and the Israel Antiquities Authority have now used architectural analysis to discover that geometry informed the layout of Göbekli Tepe’s impressive round stone structures and enormous assembly of limestone pillars, which they say were initially planned as a single structure.’

    https://phys.org/news/2020-05-geometry-earliest-temple-built-years.html

  12. tallbloke says:

    OB: Nice! The layout and photo remind me of the temples on Malta we saw a few years ago.

  13. oldbrew says:

    BCD in the expanded graphic is an equilateral triangle.

  14. Tim Spence says:

    if you (have programming skills) keep drawing lines at 137.5º to each other, you end up with a very pretty picture of a flower like a chrysanthemum, keep going and it gradually fills your screen with flowers, you just need a bit of programming to control the speed to see what is happening.

  15. tallbloke says:

    Tim, yes. If you look at the original post back in 2013 I provide a table explaining what was going on with that.

    https://tallbloke.wordpress.com/2013/09/11/tallbloke-the-why-phi-pi-slice/

    Anyway, we like pretty flowers here, so how about grabbing a few screenshots and uploading them to a picture hosting website for us to see? 🙂

  16. oldbrew says:

    222.5/137.5 divided by 2.5 is 89/55
    222.5+137.5 = 360
    360/2.5 = 144 = 89+55

    55,89,144 are Fibonacci numbers.
    https://en.wikipedia.org/wiki/Fibonacci_number

  17. tallbloke says:

    8/√φ = the number of radians in a circle to better than 1 part in a thousand.
    There are 2 π radians in a circle.

    8 ,2 and 1 are fibonacci numbers. 😉

  18. Tim Spence says:

    Well I last did that in 1996, but I’ll see if I can reproduce it in VB.net, it has some nifty drawing functions.

  19. P.A.Semi says:

    There has been a video explaining, why Phi is the most irrational number of all numbers and why the plants use it…


    (The Golden Ratio (why it is so irrational) – Numberphile)

  20. tallbloke says:

    Really good explainer, I like it.

  21. oldbrew says:

    Computing joke: there are only 10 kinds of people – those who understand binary and those who don’t.

  22. stpaulchuck says:

    Wow! and I thought I was bored and looking for something to do, ha ha ha.

  23. Paul Vaughan says:

    FYI I don’t watch any video. Period. Doesn’t matter how important. Non-negotiable.

    – – — – – — – – — – – — – – — – – — – – — – – — – – — – – — – – — – – — – – — – – —

    656.866091459218 = 1.27201964951407 * 835.546575435631
    5254.92873167374 = 8 * 656.866091459218
    5254.8227273181 (SEV)
    0.002017277483 = %error

    835.716909730743 = 657 * 1.27201964951407
    835.546575435631 (JS)
    0.020385972502 = %error

    We should fire all the MSM.
    They’re evil. Savage evil.

  24. Paul Vaughan says:

    EVERY government official ANYWHERE who suggests closing a park EVER: YOU’RE FIRED!!!

  25. Paul Vaughan says:

    Phine a11y at a 10shh!UN, attractive species 5256 — formerly 8472 but rebrand D-UN for “new normal C” — We’R serve UNdinner 4 Ant Arc TA11k Tip pin-UN luck D-own:

    S	5254.8227273181	1.618099269879	1.27204530968004	-0.00201723679013785	0.0040345956608856	0.00201727748339841
    835.546575435636	5254.92873167378	1.61803398874989	1.27201964951407	0	0	0
    N	5255.227007452	1.61785032146007	1.27194745231872	0.00567611462411994	-0.0113512627730543	-0.00567579245962905
    NUSJ	5256.25286945517	1.61721887192673	1.27169920654482	0.0251980159771098	-0.050376990151598	-0.0251916681765286
    SU	5256.57266919541	1.61702210096002	1.27162183881845	0.0312837262991866	-0.0625381046942416	-0.0312739426445643
    N-bound	5256.60566058543	1.61700180358836	1.27161385789412	0.0319115443287768	-0.0637925512513251	-0.0319013641108223
    J(SU)	5256.60603417018	1.61700157374911	1.27161376752106	0.0319186535545479	-0.0638067560980786	-0.0319084688009431
    J	5256.6393995685	1.61698104666849	1.27160569622367	0.0325535888701147	-0.0650753994494409	-0.0325429949573218
    U	5258.32377697949	1.61594529069619	1.27119836795686	0.0646068763073928	-0.12908863894265	-0.0645651627725446
    

    always differentiating the beautiful people from the ugly deep states (your country, your municipality, no matter where or who you are) — when we’re cursin’ the deep states of USA & China we’re lovin’ the Chinese and American people with all our hearts — we can write the message in right’s peak

    IT wasn’t such a subtle tipi after AI: the lost free ca11 looms write %errOors core US pond D-UN to clue luminaries too the mysterious left.

    What’s rea11y meant by “read” teams is dis-course-sieve miss story…. for new normal see.

  26. Paul Vaughan says:

    R Sci11UNs Comm. UN New Norm Hell C

    Type A Tipi for dawn a11-D: seize change 4 deep state employee recruitment formula.
    Trump R civil service compose IT shun, mixin’ 4 buoy UN C: “comm. UN at EU so CR see right now”

  27. Paul Vaughan says:

    “Give me shelter” — The Role UN Stones

    4 got the link to the recruitment formula.

  28. Paul Vaughan says:

    The STUNNING BRI11UNs of *Sharp* MISS Tory

    I couldn’t even look her in the eye.
    She was so bright.

    “pick kin up wit e-v.er’s Mayan” — Tom Petty

    “Run UN D-own a dr.eme
    Workin’ on a mess story”
    — Time Pet Tee

    Make kin a sandwich:

    657.337591300129 = 836.948215484285 * π/4
    656.866091459222 = 835.546575435636 * √Φ
    657.101756799141 = harmonic mean
    -0.000080332644 = %error
    657.101228931927 = harmonic mean
    656.236745780176 = 835.546575435636 * π/4
    657.967992714588 = 836.948215484285 * √Φ

    the last free daze
    the reign was UN stop able
    like anything was possible”

    5258.70073040103 = 836.948215484285 * 2π
    5254.92873167378 = 835.546575435636 * 8√Φ
    5256.81405439313 = harmonic mean
    -0.000080332644 = %error
    5256.80983145542 = harmonic mean
    5249.89396624141 = 835.546575435636 * 2π
    5263.7439417167 = 836.948215484285 * 8√Φ

    “there’s somethin’ Goood weight UN D-own this road”

    “hit CRews contro11
    I’m pickin’ up ‘whatever!’s my$yen”

    “We’re kin on a mystery” — Tom Petty
    Go win-ware ever IT Leads”
    “Runnin’ down a dream that never would come to me”

  29. Paul Vaughan says:

    I know I lost some of you there.

    This time we’re pitchin’ a big tent.
    It has to accommodate women and children.

    ⌊ 6.84872659292027 / 0.380883104686082 ⌉ = ⌊17.9811771870661⌉ = 18
    6.84872659292027 / 18 = 0.380484810717793
    i.e. harmonic of 6.84872659292027 nearest 0.380883104686082 is 6.84872659292027 / 18 = 0.380484810717793
    363.852449522549 = (0.380883104686082)*(0.380484810717793) / (0.380883104686082 – 0.380484810717793)

    ⌊ 363.852449522549 / 22.1392314983835 ⌉ = ⌊16.4347371113182⌉ = 16
    363.852449522549 / 16 = 22.7407780951593
    i.e. harmonic of 363.852449522549 nearest 22.1392314983835 is 363.852449522549 / 16 = 22.7407780951593
    836.948215484285 = (22.7407780951593)*(22.1392314983835) / (22.7407780951593 – 22.1392314983835)

    For reference:

    2π = 6.28318530717959
    8√Φ = 6.28921102205939

    π/4 = 0.785398163397448
    √Φ = 0.786151377757423

  30. Paul Vaughan says:

    If you’re left wondering why God’s spotlight is suddenly shining so brightly on us:

    Those are the signs that we’ve stretched the economy too far, and it’s starting to tear.”

  31. Paul Vaughan says:

    If UK woke up tomorrow and London was completely gone:
    Is everything in place to run the country soundly without London?
    Who is actually capable of replacing Trump?
    What circumstances would move China’s capital from Beijing elsewhere?
    What will it look like running the US from somewhere other than Washington?

    These are dialogues we should begin in haste.

    It’s not enough to follow a bunch of anarchists.
    We need sound back up plans.

    I always ask revolutionaries: Will there be clean, safe drinking water?
    Usually they’ve never even thought AB out how they’d assure it.

    [mod] this is way off topic

  32. Paul Vaughan says:

    Recognizing Inequality

    ….fall’s right in bet wean:


    6.28855100106194 = 33052.6240615815 / 5256
    8√Φ

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