Sorry, an error occurred while loading the content.
Browse Groups

• 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
View Source
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
Your message has been successfully submitted and would be delivered to recipients shortly.
• Changes have not been saved
Press OK to abandon changes or Cancel to continue editing
• Your browser is not supported
Kindly note that Groups does not support 7.0 or earlier versions of Internet Explorer. We recommend upgrading to the latest Internet Explorer, Google Chrome, or Firefox. If you are using IE 9 or later, make sure you turn off Compatibility View.