Look at a column in the Osireion at Abydos and you can find a pattern of overlapping circles painted on it. The same pattern turns up in places that have no known link to Egypt.

I keep asking how one shape ends up in so many hands. The sources I found disagree on almost every step of that question.

The traditional view

Mathematician George Markowsky studied the claims about the golden ratio in art and architecture. He wrote in the College Mathematics Journal that much of what is presented about it "is false or seriously misleading."

He worked from published measurements of the Great Pyramid. The slant height over half the base comes to about 1.62. He says that "differs from φ by only 0.1%." He then says it "does not appear that the Egyptians even knew of the existence of φ much less incorporated it in their buildings."

On the pyramid's perimeter, Markowsky finds a ratio within 2 percent of 2π. He asks whether that is a coincidence or whether there is a reason, and he lists several practical reasons that builders could have had for it.

The standard account of geometric signs in cave art is the one David Lewis-Williams and Thomas Dowson proposed in 1988. As Tanudirjo summarizes it, the signs come from "entoptic" patterns that the human nervous system produces in altered states of consciousness. The model rests on that nervous system being universal.

The Geometry That Keeps Showing Up

In Earthly Enigmas I treat sacred geometry as applied mathematics and not as mysticism. I follow the golden ratio through the Great Pyramid, the Parthenon, Angkor Wat, Palenque, Gothic cathedrals and Shinto gates. I write that phi sits in the Great Pyramid "to within measurable tolerance."

I write that pi is encoded to four decimal places in the Great Pyramid and that it also shows up at Teotihuacan and on Mesoamerican platforms. I say this "cannot be accidental."

The Flower of Life appears in the book on five continents across unrelated traditions. I find it carved in stone at the Temple of Osiris at Abydos. I give its date as at least 535 BCE and possibly much older. I also write that Egyptologists disagree on whether the Osireion predates Seti I and that its construction date remains genuinely uncertain.

The same forms come back in Leonardo's notebooks, in Kabbalah, in Pythagorean mathematics, in Buddhist art and in Turkish mosques. In the book I list a library of forms that includes the Fibonacci spiral, the Julia set, the vesica piscis, the Sri Yantra and the Flower of Life. I write that someone is using that library.

I follow geometric signs back about 30,000 years, on every inhabited continent. I read the entoptic explanation as an account of why the signs are universal. It does not explain to me why the motifs were preserved. I think independent invention explains simple forms. Figures with precise internal constraints must be solved, and I do not think they are stumbled into. Two cultures might both invent the wheel. I conclude that the evidence points to a source.

I correct one popular example myself. The nautilus shell follows a logarithmic spiral of about 1.33 and does not follow phi. I keep sunflowers and pine cones.

What the Geometry Leaves Open

Both views lean on questions the evidence cannot yet answer. Three of them stand out.

When was the pattern put on the column?

In the book I describe the Flower of Life at Abydos as carved, and I date it to at least 535 BCE. The sources I could reach describe it as painted.

The Archaeologist, a magazine, says that nobody knows exactly when the Flower of Life was painted at Abydos. It points to Greek text nearby that indicates sometime during the Ptolemaic era. The same article says the Osireion was first thought to be an Old Kingdom construction because of its great granite blocks. It says the structure has now been dated to the reign of Seti.

David Furlong is an author and tour leader and not an excavator. He counts at least thirteen patterns on two columns, each about 6 inches across and painted in red ochre. He reads a Christian abbreviation in letters beside them and dates the paint to perhaps as late as the fifth or sixth century CE. He first asked whether Pythagoras, whom he places in Egypt from about 535 BC, could be responsible. He writes that "Many wild claims are made for the dating of this pattern, which can now safely be dispelled."

I have not checked Furlong's reading of the letters, and I have not found an excavation report on the graffiti. The date of the Osireion and the date of the pattern are two separate questions.

When was the Flower of Life put on those columns?

Do the ratios sit in the stone?

Markowsky grants that the Great Pyramid's numbers come close. He argues that any large building offers many lengths to measure. He quotes Martin Gardner on pyramid measuring. Gardner says you will quickly have "a great abundance of lengths to play with."

Markowsky names the unsystematic hunt for the golden ratio the "Pyramidology Fallacy."

On the Parthenon he says the usual golden rectangle ignores parts of the building. He gives dimensions that produce ratios of 9/4, "well outside the acceptance range" he proposes of 1.58 to 1.66. I have read his account of those measurements and not the survey he cites.

In the book I read the same numbers as design, and I take the match as intended. Markowsky puts the pi relationship at about 2 percent. I have not checked the book's four-decimal figure against a measured survey.

What test separates a ratio the builders meant from one a measurer found?

How old is the thread?

Christopher Henshilwood and colleagues report a cross-hatched pattern drawn with an ochre crayon on a flake of silcrete at Blombos Cave in South Africa. They date it to about 73,000 years ago. They say it pre-dates the earliest previously known abstract and figurative drawings by at least 30,000 years.

That extends the book's timeline. I follow geometric signs back about 30,000 years, and the Blombos drawing sits well before that. I read the Nature abstract and not the full paper.

Tanudirjo reports that critics of the entoptic model question whether it applies everywhere. He quotes Paul Bahn asking how well established the claim of true universality is. I read Tanudirjo's account and not the 1988 paper or Bahn's comment.

Neither source says whether the signs from 73,000 years ago and the signs from 30,000 years ago belong to one tradition. Where does the thread begin?

I take up the pillar bands in Göbekli Tepe's pillars and the nested triangles in the Oregon Sri Yantra.

The questions I can't put down

Some questions need more room than one article gives them. Two of them keep pulling me back.

Which forms can a human mind arrive at alone? Lewis-Williams and Dowson rest their model on a universal nervous system. In the book I say independent invention explains simple forms and does not explain the constrained ones. I follow that line through Earthly Enigmas.

Who wrote pi in a barley field? On June 1, 2008, a formation appeared at Barbury Castle in England. Plus Maths describes it as 150 feet across and showing the first ten digits of pi, as decoded by the astrophysicist Michael Reed. Lucy Pringle doubts that humans made it, and Plus notes that it appeared on the eve of a new X-Files film. In the book I write that someone is using the library. I have not checked who made the formation. You can read a free sample before you decide. The articles section has more subjects in the same format.

Sources