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OT Pyramid puzzles

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  • john pasley
    Hi group, I wanted to experiment with pyramids and found a confusing array of material which in many ways did not stack up. Numerous geometric methods and an
    Message 1 of 1 , Jan 9, 2003
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      Hi group,

      I wanted to experiment with pyramids and found a confusing array of material
      which in many ways did not stack up. Numerous geometric methods and an
      elusive 52 degree angle persuaded me to check out the maths. This e-mail is
      important to anyone about to embark on a pyramid project or who is wondering
      why theirs doesn't 'work'. A simple overview is provided in the following
      link, for which thanks Mac:

      http://members.aol.com/kapsaris/index.html

      I identified 3 main pyramid ratio's in use (Pi, Phi and Epsilon), and a
      fourth close companion (52 degree angle) that I have included for
      completeness. Some Pythagoras and straightforward trig. gave me the
      following (though you're welcome to check the figures):

      52 DEGREE PYRAMID
      Ratio of mid base-line to apex to half base-line
      1.6243
      Angle of mid base-line to apex
      52 degrees
      Vertical height
      6.3997 cm
      Full base-line length
      10 cm
      Corner length
      9.5371 cm

      PI PYRAMID
      Ratio of mid base-line to apex to half base-line
      1.6190
      Angle of mid base-line to apex
      51.85 degrees
      Vertical height
      6.3662 cm
      Full base-line length
      10 cm
      Corner length
      9.5146 cm

      PHI PYRAMID
      Ratio of mid base-line to apex to half base-line
      1.6180
      Angle of mid base-line to apex
      51.83 degrees
      Vertical height
      6.3601 cm
      Full base-line length
      10 cm
      Corner length
      9.5106 cm

      EPSILON PYRAMID
      Ratio of mid base-line to apex to half base-line
      1.6171
      Angle of mid base-line to apex
      51.80 degrees
      Vertical height
      6.3544 cm
      Full base-line length
      10 cm
      Corner length
      9.5068 cm

      For making pyramids of any size the above figures can be turned into ratios,
      though given how close they all are it would probably take a few attempts
      before making a small pyramid of any accuracy.

      The different pyramids have been defined as follows:

      The 52 degree pyramid comes from the angle between the mid base-line length
      to the apex and the horizontal. 52 degrees is an oft quoted angle for
      pyramids but does not tie in with any of the other ratios, although many
      books claim that it does.

      The Pi formula is 4xbase-line = 2x vertical height x pi.

      The Phi pyramid has a ratio of mid base-line to apex length to 1/2 base-line
      of 1.618034 (Phi).

      All of the above three pyramids are claimed by various sources to be the
      Cheops pyramid ratios, however they are all different. Errors in determining
      which is correct arise due to the accuracy required and the disrepair of
      Cheops outer layer.

      Epsilon is another shape claiming different anomolies to those above,
      perhaps including their anomolies also. It has a base-line/apex to half
      base-line ratio of 1.6171, no reason is given for this ratio.

      Previously, I have looked at Justin's site.

      http://www.geocities.com/undergsci/pyraconstruct.html

      His cut-out shape plan (Base = 20cm, mid base-line to apex length of 16.2cm)
      lies between the 52 degree and the Pi pyramid as it has an angle of 51.88
      degrees, showing that even a tiny amount of rounding changes the ratios
      quite considerably.

      I have also seen the following formulae on this group (H/1.6171 = 1/2 B and
      B x 0.9504 = Corner length) this pyramid lies closest
      to the Epsilon pyramid as it has an angle of 51.78 degrees and a ratio of
      1.6165 It should be noted that the formulae are in some ways flawed in that
      if you start with a vertical height of 8.0855 cm, you will find Base = 10,
      Side (corner) = 9.504. By Pythagoras and trig these give a vertical height
      of 6.3503 and a ratio of 1.6165 showing that the 'H/1.6171' part of the
      formula has little bearing on the end result. You may want to consider this
      if you want a pyramid of a particular height. Perhaps using the Epsilon
      ratios above is preferable (Height is 0.63544 of base, Corner is 0.95068 of
      base)

      I also noticed the angle of base-line to apex alters with the lines starting
      position on the base-line. E.g. Epsilon has a 51.80 degree angle at mid
      base-line decreasing to 41.94 degrees at the corners. This is probably the
      cause of the required North/South alignment which is not required when using
      a cone.

      A last note is that any length or ratio I have seen given is extremely
      accurate yet invariably fails to take account of the diameter of any
      material used to make a pyramid frame. This can make a considerable
      difference, throwing your pyramid well away from any of the given ratios.
      So, the ratios use is restricted to paper full-sided pyramids. Where a frame
      is made of say copper tube you would be best off using a CAD package to
      determine the lengths giving the correct ratios.

      For those who have read this and are still awake, I hope it helps!

      John
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