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