‘Jomard?’ Talma finally prompted. ‘You don’t have to watch the fossil. It’s not going to run away.’
As if in reply, the savant suddenly took a rock hammer from his survey bag and tapped at the block’s edges. There was already a crack near the fossil and he worked with this, succeeding in splitting the nautilus specimen loose and cupping it in his hand. ‘Could it be?’ he murmured, turning the elegant creature to see its pattern in light and shadow. He seemed to have forgotten our mission, and us.
‘We’ve still a way to go to the top,’ I warned, ‘and the day is getting late.’
‘Yes, yes.’ He blinked as if waking from a dream. ‘Let me think about this up there.’ He put the shell in his satchel. ‘Gage, hold the tape. Talma, ready your pencil!’
The summit took another half hour of careful climbing. It was more than 450 feet high, our measuring showed, but we could produce no more than a rough approximation. I looked down. The few French soldiers and Bedouins we could see looked like ants. Fortunately the pyramid’s capstone was gone, so there was a space about the size of a bed on which to stand.
I did feel closer to heaven. There were no competing hills, just flat desert, the winding silver thread of the Nile, and the collar of green on each of its shores. Cairo across the river shimmered with a thousand minarets, and we could hear the wail of the faithful being called to prayer. The battlefield of Imbaba was a dusty arena, dotted with pits where the dead were being tossed. Far to the north, the Mediterranean was invisible over the horizon.
Jomard took out his stone nautilus again. ‘There is clarity up here, don’t you think? This temple focuses it.’ Plopping down, he began to jot some figures.
‘And not much else,’ Talma said, sitting himself in exaggerated resignation. ‘Did I mention that I’m hungry?’
But Jomard was lost again in some world of his own, so finally we were quiet for a while, having become accustomed to such meditation by the savants. I felt I could see our planet’s curve, and then scolded myself that it was illusion at this modest height. There did seem a kind of benign focus at the structure’s summit, however, and I actually enjoyed our quiet isolation. Had any other American been up here?
Finally Jomard abruptly rose, picked up a limestone fragment as big as his fist, and hurled it as far as he could. We watched the parabola of its fall, wondering if he could throw far enough to clear the pyramid’s base. He couldn’t, and the stone bounced off the pyramid’s stone blocks below, shattering. Its pieces rattled down.
He looked down the slope for a moment, as if considering his aim. Then he turned to us. ‘But of course! It’s so obvious. And your eye, Gage, has been the key!’
I perked up. ‘It has?’
‘What a marvel we are standing on! What a culmination of thought, philosophy, and calculation! It was the nautilus that let me see it!’
Talma was rolling his eyes.
‘Let you see what?’
‘Now, has either of you heard of the Fibonacci sequence of numbers?’
Our silence was answer enough.
‘It was brought to Europe about 1200 by Leonardo of Pisa, also known as Fibonacci, after he had studied in Egypt. Its real origin goes much further back than that, to times unknown. Look.’ He showed us his paper. On it was written a series of numbers: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55. ‘Do you see the pattern?’
‘I think I tried that one in the lottery,’ Talma said. ‘It lost.’
‘No, see how it works?’ the savant insisted. ‘Each number is the sum of the two before it. The next in the sequence, adding 34 and the 55, would be 89.’
‘Fascinating,’ Talma said.
‘Now the amazing thing about this series is that with geometry, you can represent this sequence not as numbers but as a geometric pattern. You do so by drawing squares.’ He drew two small squares side by side and put a number 1 inside each. ‘See, here we have the first two numbers of the sequence. Now we draw a third square alongside the first two, making it as long as they are combined, and label it number 2. Then a square with sides as long as a number 1 square and number 2 square combined, and label it 3. See?’ He was sketching quickly. ‘The side of the new square is the sum of the two squares before it, just as the number in a Fibonacci sequence is the sum of the two numbers before. The squares rapidly get bigger in area.’
Soon he had a picture like this:
‘What does that number at the top, the 1.6 something, mean?’ I asked.
‘It’s the proportion of the length of the side of each of the squares to the smaller one before it,’ Jomard replied. ‘Notice that the lines of the square labelled 3 have a proportional length with the lines of the squares 2 as, say, the proportion between square 8 and square 13.’
‘I don’t understand.’
‘See how the line at the top of square 3 is divided into two unequal lengths by its junction with squares 1 and 2?’ Jomard said patiently. ‘That proportion between the length of the short line and the long line is repeated again and again, no matter how big you draw this diagram. The longer line is not 1.5 times the length of the shorter, but 1.618, or what the Greeks and Italians called the golden number, or golden section.’
Both Talma and I straightened slightly. ‘You mean there’s gold here?’
‘No, you cretins.’ He shook his head in mock disgust. ‘Only that the proportions seem perfect when applied to architecture, or to monuments like this pyramid. There’s something about that ratio which is instinctively pleasing to the eye. Cathedrals were built to reflect such divine numbers. Renaissance painters divided their canvases into rectangles and triangles echoing the golden section to achieve harmonious composition. Greek and Roman architects used it in temples and palaces. Now, we must confirm my guess with measurements more precise than those we’ve made today, but my hunch is that this pyramid is sloped precisely to represent this golden number, 1.618.’
‘What has the nautilus to do with anything?’
‘I’m coming to that. First, imagine a line descending under our feet from the tip of this colossus to its base, straight down to the desert bedrock.’
‘I can confirm it is a long line, after that hard climb up,’ Talma said.
‘More than four hundred and fifty feet,’ Jomard agreed. ‘Now imagine a line from the centre of the pyramid to its outside edge.’
‘That would be half the width of its base,’ I ventured, feeling the same two steps behind that I’d always felt with Benjamin Franklin.
‘Precisely!’ Jomard cried. ‘You have an instinct for mathematics, Gage! Now, imagine a line running from that outside edge up the slope of the pyramid to where we are here, completing a right triangle. My theory is that if our line at the pyramid’s base is set as one, such a line up to the peak here would be 1.618 – the same harmonious proportion as shown by the squares I’ve drawn!’ He looked triumphant.
We looked blank.
‘Don’t you see? This pyramid was built to conform to the Fibonacci numbers, the Fibonacci squares, the golden number that artists have always found harmonious. It doesn’t just feel right to us, it is right!’
Talma looked across to the other two large pyramids that were our neighbours. ‘So are they all like that?’
Jomard shook his head. ‘No. This one is special, I suspect. It is a book, trying to tell us something. It is unique for a reason I don’t yet understand.’
‘I’m sorry, Jomard,’ the journalist said. ‘I’m happy for you that you are excited, but the fact that imaginary lines equal 1.6, or whatever you said, seems an even sillier reason to build a pyramid than calling something pointed a hemisphere or building a tomb you won’t be buried in. It seems perfectly possible to me that if any of this is true, your ancient Egyptians were at least as crazy as they were clever.’
‘Ah, but that is where you are wrong, my friend,’ the savant happily replied. ‘I don’t blame your scepticism, however, because I didn’t see what has been staring us in the face all day until sharp-eyed Gage here helped me find the fossil nautilus. You see, the Fibonacci sequence, translated into Fibonacci geometry, yields one of the most beautiful designs in all nature. Let’s draw an arc through these squares, from one corner to another, and then connect the arcs.’ He flipped his drawing. ‘Then we get a picture like this:’