Jim Eagles thinks he has the answer to the riddle of the Sphinx's enigmatic grin.
It's hardly surprising that the Sphinx's huge shattered face seems to smile enigmatically. The gateway to one of the world's most famous monuments is marked by a handprinted cardboard sign proclaiming: Entrance.
The millions of tourists who make the pilgrimage here every year must pass under that sign, and through a gap in the rickety metal and wire fence, to come face to face with the giant figure of a lion with a man's face. Inside the fence, souvenir sellers fight over positions, tourist police drag children off by the ears for trying to sell carved camels to tourists, and rubbish lies in the corners.
It's hard to think of a less appropriate setting for the figure the Arabs call the Father of Terror. No wonder he is amused.
But it doesn't seem to matter.
No matter how many times you've seen the Sphinx on television it is still awe-inspiring to be in the presence of this extraordinary figure carved 4600 years ago from a huge block of limestone left over from the building of the Great Pyramid of Cheops.
To add to that awe, behind the sphinx's battered face looms the vast shape of the great pyramid itself, and the smaller attendant pyramids of Chephren and Mycerinus.
The effort required to carve the sphinx, 50m long and 22m high, seems remarkable enough. But the act of building the great pyramid seems beyond belief. The base blocks stand nearly as tall as my wife and are said to weigh 15 tonnes. Yet this massive structure is estimated to contain 2.5 million blocks, each of which had to be hauled into place.
No photo or documentary can adequately convey its overpowering presence.
Like the sphinx, it is majestic enough to overcome the petty grubbiness of its setting: more rubbish, howling vendors, scavenging kids and a hundred stinking buses and vans.
It is so big - 137m high with its base covering five hectares - that you barely notice the thousands of other tourists who have come to pay their respects to the oldest tourist attractions on the planet.
If you have the energy you'll get a better feel for its size if you walk the 1km around the base.
Failing that, you can take a camel or horse ride, although any prospect we might do that was shattered by the number of horror stories of tourists being led over the sand dunes and then subjected to blackmail ... or worse.
It's certainly worth carrying on to the designated viewing area for the extraordinary contrast between the three pyramids in the desert sand and the vast, smoggy sprawl of Cairo pressing in on three sides.
Unless you're very enthusiastic you probably can't get inside the Pyramid of Cheops - they sell only 180 tickets a day - but there is apparently little to see.
You can go into the Pyramid of Mycerinus, but it's a claustrophobic experience to be crammed with hundreds of pushing, sweaty tourists into a narrow stone passage littered with drink cans and smelling of a million unwashed bodies and what seems suspiciously like stale urine. If you want to see the inside of a tomb it's better to wait for Luxor and the Valley of the Kings.
It's easy to see why the later pharaohs thought - mistakenly as it turns out - that their tombs would be safer in the valley than inside a pyramid because it is one of the most forbidding and inhospitable places on earth.
But into its bare rock over a period of several hundred years a vast workforce dug a series of giant mausoleums that are still stunning thousands of years later.
Their tunnels go as far as 100m into solid rock, with storage chambers along the way and burial chambers at the end, all carefully plastered, carved with figures and then painted.
A ticket gives entrance to three tombs and, although all are different, it doesn't matter greatly which you choose because they are all extraordinary.
The treasures they once contained have long since been stolen or carried off to museums, although a few still contain a giant sarcophagus and in one or two cases the mummy of some lesser individual.
But all are rich with paintings depicting the wisdom, virtue, power and godliness of the pharaohs whose bodies they were built to house.
It feels incredible to be able to walk down these steamy, smelly tunnels and see paintings dating back 3500 years.
In the tomb of Ramses III, one side chamber has beautiful pictures showing musicians playing for the pharoah and his fellow gods.
In another the artists have depicted all the foods, utensils, furnishings and other goods - once stored there - that he would need in the afterlife.
In yet another are poignant scenes of everyday life in Egypt. They don't show people demanding backsheesh but I suspect it happened back then because it seems bred into the local psyche.
The guards in the tombs want payment for offering unnecessary directions, for handing out pieces of cardboard able to be used as fans - and for just being there.
Their attitude was summed up for me in a delightful incident when one guard caught a visitor breaking the ban on photography.
Instead of looking cross he emitted a satisfied "Aaaaaaah," broke into a beatific smile and held his hand out. The photo cost E5 (about $1.25).
You can't help wondering what the pharaohs would have felt about the fate of these resting places into which they put so much effort.
Some of them were promptly taken over by their successors with names changed and portraits altered. Most were fairly quickly plundered by grave robbers.
The rest have since been stripped by modern relic hunters and museums. Many of the surviving figures were disfigured by Coptic Christians, who occupied many tombs and temples 1800 years ago to demonstrate that the pharoahs were not real gods.
And now their tombs are invaded by hordes of unwashed common people and turned into a venue for petty blackmail.
But the tombs have certainly given their builders a form of immortality. After all, which of our modern pharaohs and how much of our modern building will still be remembered in even 1000 years?
The mummification process has also, in a way, allowed the pharaohs to defy the centuries by making it possible for the bodies of these self-appointed gods to be seen at the Egyptian Museum in Cairo.
There you can discover that the most megalomaniac builder of them all, Ramses II, was a big man in every sense.
Egypt is full of his images, including the 12m fallen statue at Memphis, the two Colossi of Memnon which are 17m high, and the astounding statues which welcome visitors to Abu Simbel, which stand more than 20m.
All that building obviously wasn't because he suffered from an inferiority complex because his mummy is well over 1.8m tall and even in its shrunken state exudes power.
The other highlight at the museum is the magnificent display of treasure recovered from the tomb of Tutankhamen, including his stunning gold death mask.
It's a bit ironical that King Tut was an insignificant teenage pharaoh whose only claim to fame is that his tomb escaped being robbed because its entrance was covered by debris from the excavation of tombs for his more powerful successors.
If the belongings he took to the afterlife were as fabulous as this, I can't help wondering what Ramses II would have needed.
At the Valley of the Kings, I was particularly intrigued to visit the tomb of Ramses III, partly because of the extraordinary vividness of its paintings but also because it was here that the first recorded strike in history took place.
Apparently the food with which the workers were paid failed to arrive for three months and they stopped work until Ramses forked out.
After that, it's well worthwhile going to what is now called the Valley of the Workers, over the hill from the Valley of the Kings, to see how these talented artisans lived and died.
They evidently worked in two teams, on alternate nine-day shifts, and on their days off spent times with their families in a purpose-built village.
The walls of their tidy little three-roomed state houses are still standing, together with some of the community facilities they enjoyed.
Even more fascinating are the tombs of two supervisors. Although much smaller than those of the pharaohs, they are possibly even better painted.
These show the workers' idea of heaven, depicting them lying back while musicians play, being waited on hand and foot and enjoying the finest of foods.
They tell a far more down-to-earth story than those featuring the pharaohs, whose paintings are simply propaganda designed to show them as gods who must be obeyed.
One of the supreme examples of that is the incredible Temple of Horus at Edfu.
It is a mere 2000 years old, but was built by the Ptolemys in a copy of the style of the first temples 2000 years earlier.
The Ptolemys, descended from one of Alexander the Great's generals, tried to ingratiate themselves with the Egyptians by adopting their heritage.
The murals on the walls show these Greek pharaohs dressed in traditional clothing, smiting their enemies while holding them by the hair, hobnobbing with gods, and being thanked by the mighty falcon god Horus for rebuilding his temple. Most of the other temples also have paintings and statues designed to show the pharaohs being treated as equals by the gods, and in some cases being fathered by them, no doubt to encourage the masses to show respect. Spin-doctors are definitely not a modern phenomenon.
As many travel writers have commented, it is impossible to find words to convey the impact of these magnificent buildings, their soaring pillars, towering walls, spacious courtyards and intimate sanctuaries.
Look closer and you find all of them covered in superb engravings of pharaohs, gods, men and their belongings, animals, and their stories.
Look higher and you often find patches of faded colour which show that all these pictures had been painted in brilliant colours. Study the doorways and you will see where they were once closed with huge wooden doors, often covered with gold. Peer into the sanctuaries and remember that they were built to house magnificent statues of the gods with their animal heads and human bodies.
Gaze in amazement at the giant statues of pharaohs, 10, 15, 20m high, and remember that there used to be many more.
Then try to imagine the impact all this had on the humble peasants. No wonder they thought the pharaohs were gods. No wonder, even, that the pharaohs came to think they really were gods.
You can get an idea of the extraordinary resources which went into these marvels from the cost of repairing them.
The Egyptian Government has restored the foundations of just 20 columns at the magnificent Luxor Temple. It took three years and cost E20 million ($5 million). So just imagine the resources it took to build them in the first place. And, for that matter, the skill.
The inner sanctuary at Abu Simbel was designed so that twice a year - believed to be on the anniversaries of Ramses II's birth and coronation - the rays of the rising sun illuminated the statues of himself and two fellow gods.
When it was moved to escape the rising waters of Lake Nasser modern engineers were unable to calculate the positions quite so accurately and the anniversaries are now marked a day late.
Standing amid these magnificent buildings it is hard to believe ordinary mortals created them all those thousands of years ago with nothing more than muscle-power and a knowledge of mechanics - especially when today it seems to be beyond the wit of man to come up with a proper entrance sign.
No wonder some people seriously believe the temples and pyramids are the work of creatures from outer space ... and no wonder the sphinx still seems to be smirking.
Visas: New Zealand passport holders need a visa to visit Egypt but three-month tourist visas can be bought on arrival for US$15 ($22).
Currency: The Egyptian currency is a pound (or guinay in Arabic). You get about 4 for NZ$1. Egyptian currency is not obtainable outside the country, but United States dollars, travellers' cheques and credit cards are widely used.
Tipping: This custom is a part of Egyptian life. Make sure you keep a supply of small-denomination notes to deal with the inevitable demands from airport representatives, drivers, porters and guides.
Getting there: United Travel has return flights from Auckland to Cairo.
Getting around: Ancient Kingdoms Holidays has a fantastic Jewels of the Nile tour available through United Travel. The nine-day itinerary starts and ends in Cairo and includes a five-day Nile cruise.
Jim Eagles and Alan Gibson travelled as guests of United Travel and Ancient Kingdoms Holidays.