The Law of Large Numberless Caves

In Xanadu did Kubla Khan
A stately pleasure-dome decree:
Where Alph, the sacred river, ran
Through caverns measureless to man
Down to a sunless sea.

Samuel Taylor Coleridge, Kubla Khan (1797-8)

Coleridge’s river Alph may have been the fantastical residue of an opium haze but there is something we can say about it that remains solidly true for all time: if we could indeed measure its measureless run from source to sunless sea and divide its length by the distance as the proto-Castanedan-crow flies, we would reach a good approximation for the number pi (roughly 3.14159). The ‘more’ measureless the caverns the better the approximation.

Coleridge wouldn’t have known this at the time but today there is no doubt we can confidently assign this numerical property to his dreamed motif. The relationship between pi and river ratios was discovered by Cambridge earth scientist Hans-Henrik Stolum in 1996. Writer Simon Singh, best-selling author of Fermat’s Last Theorem describes Stolum’s research in the same book to illustrate the rich connections which exist between nature and number. Some of these connections, like those found by Stolum, reveal themselves only when there is a large selection of data to work with. And the larger the selection the greater the precision.

From ephemeral visions to plausible fictions: in the US in 1885 a pamphlet was published in Lynchburg, Virginia, recounting the intriguing tale of one Thomas J. Beale and a hidden fortune in buried treasure. Entitled The Beale Papers it described how, in 1822, Beale entrusted a box containing three papers to hotelier Robert Morriss of the Washington Hotel. Beale subsequently disappeared and when Morriss opened the box he found that each page was a numerically encrypted ciphertext.

The story goes that Morriss wasted his remaining years attempting to decipher the papers until he passed them onto an anonymous friend who later succeeded in cracking the second page after he realised it had been encoded using letter position numbers from the Declaration Of Independence. This second page, apparently written by Beale, described how he had cached around three tons of gold, silver, and jewels in an excavated vault in Bedford County, Virginia. It also tantalisingly informed the reader that the first sheet detailed the location of the treasure, and that the third gave the names of Beale’s prospector companions, who had an equal claim to the hoard.
To date the treasure has never been found, and The Beale Papers is considered by many modern cryptanalysts to be a work of pure fiction.

The strength of their conviction may well be supported by a statistical analysis carried out by US computer scientist James Gillogly in 1980. Gillogly applied the same encryption rule for paper 2 to paper 1 and found sequences of consecutive letters of the alphabet, such as “ABFDEFGHIIJKLMMNOHPP” which suggested a lazy, yet deliberate, attempt to produce additional random-looking ciphertext from the DOI.

We can but dream of the scale of ‘caverns measureless to man’ that have been scoured throughout Bedford County and beyond in the hope of recovering a haul worth in excess of 60 million dollars in today’s money! Seeing as it has still eluded the best minds and miners for over a century, it’s a safe bet that it never existed.

From plausible fictions to credible facts: in 2006 a cave in the Tsodilo Hills of Botswana hit the scientific headlines after archaeologists discovered a 20 foot python head, carved from stone, inside. Nearby they unearthed spearheads which had been ritually burnt, suggesting humans had developed the capacity for ritual practice at least 30,000 years earlier than the currently accepted estimate of 40,000 years ago.

They also came across the entrance to a hidden chamber behind the python which revealed tell-tail signs of habitation. University of Oslo scientist Sheila Coulson, a member of the team, theorized that ancient shamans would hide alone inside the secret chamber before speaking to those who gathered round to burn their offerings, thus making it appear as though the python was talking directly to them.

The shaman’s cave embodies a mathematical abstraction that remained hidden for many millennia until its discovery in the 20th century: that of the Unilluminable Room. Prolific polymath Clifford Pickover in The Math Book relates that this concept was first formulated by mathematician Ernst Strauss in the 1950s. The problem asks whether it is possible for mirrored rooms of certain curvature to exist such that light from a point source, like a match (or a fire), will always fail to fully illuminate their interiors.

The proof that such rooms could exist came in 1958 when physicist and topologist Roger Penrose mathematically constructed a room that had dark areas no matter where the light source was placed. Since then more rooms have been constructed, both curved and polygonal, which cannot be fully lit by any number of light sources.

Perhaps the Tsodilo shaman’s secret chamber was just such a room; a fortuitously hewn, geological godsend ready to light the touchpaper for an explosive emergence of superstition? Or perhaps – given the predominance of ancient caves and communities – the totemistic mystification of the landscape was inevitable?

The ‘Pi-shaped’ Alph, the alphabet encryptions, the shamans in the shadows: what could they have in common? It appears the law of expectation, but who would have thought that?

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