Bob, I couldn’t let this elegant and simple solution languish unanswered. I’ve just tried it, and it works amazingly well and is very accurate – even done by an amateur such as me, new to the world of sacred geometry. I would only add one slight tweak, which means this can be done without even using a scale ruler .. i.e. do it the old school way, with just a straight edge and compass, utilising just proportions.

Having drawn the first small (4 in this instance) circle centrally on the initial horizontal line, set a compass to its diameter and draw the first large circle round it at twice the size (8 in this instance). Where the edge of this large circle intersects the initial horizontal line (either side will do), draw a vesica with the small (4) radius. Draw a vertical line between these two intersection points on the double size circle, and where it cuts the horizontal is 1/4 of the small circle radius, .. so you now have a new scale and radius from the centre point for the big circle, in the proportions 7:4 rather than 8:4. This whole method will now scale to any size, with any random initial radius.

(I wasn’t sure what function your last two small circles performed, I managed to draw this successfully without them. I’m probably missing something…)

The elegance of your solution suggests, to me at least, that ancient Egypt not only understood both Pi (a close approximation of) and Phi, the Kepler triangle (and therefore also what became Pythagoras’ theorem); but in this design and its subsequent realisation as Khufu’s pyramid, elegantly tied them all together. It’s quite beautiful.

Of course, Wikipedia is sure that any connection between Khufu’s monument and Kepler’s triangle is purely coincidental; but I think more and more people are getting wise to this nonsense.

…and a huge hat tip to this site, Gary, your work is superb. Thank you. I’ve just found this place, I’ll be spending a lot of time here.

From the above link:
“1) Draw horizontal and vertical lines (axes) from the pyramid centers.

2) Draw an exact golden diagonal from the intersection of ‘d’ and ‘e’.
(Any rectangle with this diagonal is a true golden rectangle.)

3) Draw lines ‘h’ and ‘g’ from the intersection of ‘c’ and ‘f’.
Now, the rectangle formed by lines a-b-c-h is an extraordinarily accurate golden rectangle:

Instead of 74 inches, the “inaccuracy” of its long side now stands at 0.18 inches over the distance of 17,833.8 inches ..
The ratio of sides in this golden rectangle is 1.618021, which is just 0.00001 off the true value of Phi.

The line ‘g’ is an extraordinarily accurate golden diagonal:
Instead of the correct 31.71747.. degrees, its angle is 31.71767.. mere 2/10,000 (half a second) degree off.”

Interestingly I think both the author of the above article and Bauval are correct. The unification lies in the disparities

For example:

175.2 − 172.8 = 2.4
175.2 − 172.6 = 2.6
1,728 ÷ 216 = 8

172.6 ÷ 216 × 864 = 690.4
690.4 − 230.4 = 460
460 ÷ 2 = 230

175.2 ÷ 21.6 × 864 = 7,008
172.8 ÷ 216 × 864 = 691.2

7,008 + 691.2 + 690.4 ÷ 9.6 × 864 = 755,064
755,064 ÷ 64 ÷ 0.875 × 7 = 94,383
94,383 ÷ 6.40791100124 ÷ 1,618 × 9 = 81.9296875
9 × 2 × 9 × 6 × 8 × 7 × 5 = 272,160
272,160 ÷ 21 = 12,960 (half Plato’s wheel [Great Year] of 25920)

6.40791100124 is a ratio I discovered that pulls the phi element out of the harmonic numbers,
I therefore dubbed it the Magic Ratio. You can simplify the above – I left it as the
expanded version for you to play.

216 ÷ 81 × 9 = 24

81 × 9 = 729

A friend of mine used this configuration to build a tesla ‘donut’ – it works perfectly – lightning shows and conductor arc sprays. Have pics if you want to email for them.

I’ll leave you with a couple of examples of how my Magic Ratio works:
729 ÷ 6.40791100124 ÷ 161.8 × 144 = 101.25
729 ÷ 101.25 = 7.2

If you have familiarity with the harmonic numbers the sequencing will be immediately identifiable.

Another key lies in: 7.2 × 729 ÷ 1,618 ÷ 6.40791100124 = 0.50625
0.50625 × 6.666666666666 = 3.375

Those familiar with Dürer’s l’elemocina (Melencolia), Plato and John MIchell’s Dimensions of Paradise, should recognise the significance of that sequence.

In ancient times much knowledge was transferred orally rather than in written formats. Some of it may also have been guarded as a secret of the trades that used it.

There is good source material for “deeper” analysis of the early roots of the Golden Ratio but you have to be willing to do some digging.

https://www.researchgate.net/publication/269709899_Follow_the_Golden_Ratio_from_Africa_to_the_Bauhaus_for_a_Cross-Cultural_Aesthetic_for_Images

This concept of excluding the contributions of African, Asian, Latin, and Middle Eastern mathematicians is deep and ugly and clearly methodical. The so called Pacal’s triangle was not discovered by Blaise Pascal (Persia and China both used the triangle hundreds of years before him but again this is excluded from traditional education in the Western World!!! All I ask is why? The GR is no different…How can something be used throughout entire countries and incorporated into art, fashion, religion, and architecture literally thousands of years before Greece existed or Italy and it not be taught as originating someplace else and being “formalized” by the Greek greats…I think you are correct that it is not all racism but rather a disagreement on human usage of math and science. The Nubians, Egyptians, Babylonians and others in the ancient world saw math and science as a religious and practical tool vs science for study in and of itself. That makes the Greeks very different as they studied math for formalization and axiomatic representation over practical application as their predecessors did. In today’s time add the ugliness of racism with a different approach and you get total exclusion of the originators which brings us to today where most people have NO connection to advanced math and those that do do not have the historical connection to piece it all together. Folks like ourselves have to help make it connect to people’s lives and showcase math connects to everyone (all ethnicities). A start for your excellent website is starting a subsection on the multicultural connection of the GR… This would be huge!!!!! Let me know if I can help!!!!

I agree entirely that we should make the history and applications of knowledge as complete, accurate and inclusive as possible. My book (https://bit.ly/goldenratiobook), published in 2018, includes contributions of Al-Khwarizmi and Abu Kamil to mathematics. My section on human facial beauty illustrates how the golden ratio in human facial proportions is found in all races/ethnicities, and provides examples of Caucasian, Burmese, Brazilian, South Indian, Lebanese and South African models to illustrate the point. I believe it is important to relate to people across all races, ethnicities and cultures to bring them together, and to help us to see that we are all one. I think this is best done in a very positive, inclusive way rather than driving division with an emotionally charged focus on bias, racism, etc.

I love your platform and the research you have done!!! BZ! I just think you need to expand it for the 21st century and include new research that is multicultural, exciting, and engaging for future generations. I invite you to check out some of the research of Dr Ron Eglash on African Fractals which connects to a wider audience. I also think the spiritual connection is an important one as The GR is a universal connector to whatever we call God as the divine Creator of the universe. Maybe even get some of the faith leaders to learn some math to go along with their preaching from churches, synagogues, and mosques…What a powerful connection to marry faith and mathematics!!!