Last Fractal
Ok, here’s a last fractal (for now). I’m just completing a set of fractal images for printing 6′x6′ (that’s big!) and will finish off this month’s blog by showing you just one more:
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| Detail within the Mandelbrot set fractal. This detail is found by zooming in on the overall Mandelbrot set image, finding edges and buds with interesting features. Fractals are complex geometric shapes that exhibit repeating patterns typified by self-similarity, or the tendency for the details of a shape to appear similar to the shape itself. Often these shapes resemble patterns occurring naturally in the physical world, such as spiraling leaves, seemingly random coastlines, erosion and liquid waves. Fractals are generated through surprisingly simple underlying mathematical expressions, producing subtle and surprising patterns. The basic iterative expression for the Mandelbrot set is z = z-squared + c, operating in the complex (real, imaginary) number set. Image: 10376 Species: Mandelbrot set |
Fractal of the Day
Whoa, here’s another crazy fractal from the Mandelbrot set. When I was setting the colors on this one it sort of reminded me of the trippy hippy art of the ’70’s. Right on man.
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| The Mandelbrot Fractal. Fractals are complex geometric shapes that exhibit repeating patterns typified by self-similarity, or the tendency for the details of a shape to appear similar to the shape itself. Often these shapes resemble patterns occurring naturally in the physical world, such as spiraling leaves, seemingly random coastlines, erosion and liquid waves. Fractals are generated through surprisingly simple underlying mathematical expressions, producing subtle and surprising patterns. The basic iterative expression for the Mandelbrot set is z = z-squared + c, operating in the complex (real, imaginary) number set. Image: 18737 Species: Mandelbrot set |
Mandelbrot Fractal Picture
Fractal ‘o’ the day, from the classic Mandelbrot family of inequalities:
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| Detail within the Mandelbrot set fractal. This detail is found by zooming in on the overall Mandelbrot set image, finding edges and buds with interesting features. Fractals are complex geometric shapes that exhibit repeating patterns typified by self-similarity, or the tendency for the details of a shape to appear similar to the shape itself. Often these shapes resemble patterns occurring naturally in the physical world, such as spiraling leaves, seemingly random coastlines, erosion and liquid waves. Fractals are generated through surprisingly simple underlying mathematical expressions, producing subtle and surprising patterns. The basic iterative expression for the Mandelbrot set is z = z-squared + c, operating in the complex (real, imaginary) number set. Image: 10378 Species: Mandelbrot set |
More fractal pictures.
Julia Set Fractal
Today’s fractal picture is:
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| Fractal designl. Fractals are complex geometric shapes that exhibit repeating patterns typified by self-similarity, or the tendency for the details of a shape to appear similar to the shape itself. Often these shapes resemble patterns occurring naturally in the physical world, such as spiraling leaves, seemingly random coastlines, erosion and liquid waves. Fractals are generated through surprisingly simple underlying mathematical expressions, producing subtle and surprising patterns. The basic iterative expression for the Mandelbrot set is z = z-squared + c, operating in the complex (real, imaginary) number set. Image: 18732 Species: Mandelbrot set |
I think this one was generated with from the Julia set rather than the Mandelbrot set, but not 100% sure.
Fractal Picture
Here’s another example of detail within the Mandelbrot fractal set, illustrated by assigning color to the number of iterations required for the expression to explode:
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| The Mandelbrot Fractal. Fractals are complex geometric shapes that exhibit repeating patterns typified by self-similarity, or the tendency for the details of a shape to appear similar to the shape itself. Often these shapes resemble patterns occurring naturally in the physical world, such as spiraling leaves, seemingly random coastlines, erosion and liquid waves. Fractals are generated through surprisingly simple underlying mathematical expressions, producing subtle and surprising patterns. The basic iterative expression for the Mandelbrot set is z = z-squared + c, operating in the complex (real, imaginary) number set. Image: 18729 Species: Mandelbrot set |
Photo of the Temple of Hatshepsut
The Temple of Hatshepsut is the mortuary temple for Pharoah Hatshepsut, one of the rulers of the 18th Dynasty of ancient Egypt and one of the most successful female pharoahs in history. The Temple of Hatshepsut is located at Deir el-Bahri, near the Valley of the Kings, across the Nile River from Luxor, Egypt. The colonnaded Djeser-Djeseru building, shown here, lies at the foot of spectacular cliffs and is worth a visit when travelling in the Nile Valley.
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| Temple of Hatshepsut. Luxor, Egypt. Image: 02588 Location: Luxor, Egypt View this Image in Google Earth! |
Photo of the Ramesseum
The Ramesseum is the mortuary temple for one of the greatest rulers the world has ever known: Pharoah Ramesses II. Also known as Ramesses the Great, Ramesses II was a 19th Dynasty pharaoh who ruled for 67 years during the 13th century B.C. The Ramesseum is part of the ancient Theban necropolis, across the Nile from what is now Luxor, Egypt. The ironic sonnet Ozymandius aluded to Ramesses II.
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| Ramesseum. Luxor, Egypt. Image: 02586 Location: Luxor, Egypt View this Image in Google Earth! |
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| Ramesseum. Luxor, Egypt. Image: 02590 Location: Luxor, Egypt View this Image in Google Earth! |
Photo of the Borrego Badlands from Fonts Point
From Font’s Point in Anza-Borrego Desert State Park, one can look across the Borrego Badlands (Carrizo Badlands), once an ancient lake but now a corrugated, tortured, barren wasteland (e.g., Mordor).
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| Carrizo Badlands viewed from Fonts Point. Anza-Borrego Desert State Park, Anza Borrego, California, USA. Image: 05530 Location: Anza-Borrego Desert State Park, Anza Borrego, California, USA View this Image in Google Earth! |
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| Carrizo Badlands viewed from Fonts Point. Anza-Borrego Desert State Park, Anza Borrego, California, USA. Image: 05533 Location: Anza-Borrego Desert State Park, Anza Borrego, California, USA View this Image in Google Earth! |
Font’s Point photos, Anza-Borrego Desert State Park photos, Borrego Badlands photos, Carrizo Badlands.
Photo of 17 Palms Oasis
17 Palms Oasis is located in Anza-Borrego Desert State Park, several miles down the lunar-looking Arroyo Salado wash. The drive along Arroyo Salado is fun and relatively simple, although a four-wheel drive car is recommended. Seventeen Palms Oasis, consisting of (not surprisingly) 17 palms, appears out of nowhere as you drive down the wash and is a curious finding in the middle of such a barren landscape. It exists because of a shallow, subsurface supply of fresh water.
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| Seventeen Palms Oasis, Borrego Badlands. Anza-Borrego Desert State Park, Anza Borrego, California, USA. Image: 05538 Location: Anza-Borrego Desert State Park, Anza Borrego, California, USA View this Image in Google Earth! |
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| Seventeen Palms Oasis, Borrego Badlands. Anza-Borrego Desert State Park, Anza Borrego, California, USA. Image: 05539 Location: Anza-Borrego Desert State Park, Anza Borrego, California, USA View this Image in Google Earth! |
17 Palms Oasis photos, Seventeen Palms Oasis photos, Anza-Borrego Desert State Park photos.
Sunset Colors
Over the years I have had the good fortune to spend many weeks on liveaboard boats, in remote locations, diving and shooting photos. Typical trips will schedule some night dives, however we are not hard core on the diving and usually prefer to stay dry in favor of happy hour and an end-of-day skiff ride along the shore. Sunset on these trips is a golden time: gather on the sundeck, shake the last bits of seawater water out of the ears, enjoy one of Bob Heil’s blender concoctions and watch the sky light up.
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| Clouds and sunlight. La Jolla, California, USA. Image: 04817 Location: La Jolla, California, USA |
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| Sunset and water, Sea of Cortez. Image: 00285 |
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| Clouds and sunlight. Maui, Hawaii, USA. Image: 05642 Location: Maui, Hawaii, USA |
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| Clouds at sunset, rich warm colors and patterns. Guadalupe Island (Isla Guadalupe), Baja California, Mexico. Image: 18514 Location: Guadalupe Island (Isla Guadalupe), Baja California, Mexico View this Image in Google Earth! |
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| Clouds at sunset, rich warm colors and patterns. San Clemente Island, California, USA. Image: 18515 Location: San Clemente Island, California, USA View this Image in Google Earth! |
Underwater Light
Much of my time underwater is spent looking for simple, available light photographs. These are not sexy photos, but they tend to sell well and are typically used as backgrounds or screened back with text or inset photos laid over the top. The best part is they are simple to shoot, even for me.
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| The ocean surface, seen from underwater, ripples with waves and wind and bright sunlight. Bahamas. Image: 00700 Location: Bahamas |
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| Water patterns. Image: 03181 |
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| Sunlight spreads across broad sand plains, trochoidal patterns. Bahamas. Image: 03185 Location: Bahamas |
Photo of the Egyptian Pyramids
We wrapped up our stay in Egypt with some time at my brother’s house in Cairo, touring the older parts of the city and seeing some off-the-beaten-path markets, mosques, and other amazing stuff. One highlight of our time in Cairo was a day at the Giza necropololis to see the Great Pyramids, and another day in Saqqara to see very different pyramids. Tracy is seen below riding a camel on the sand plains of Giza, with the Great Pyramids in the distance. Seeing one of the Seven Wonders of the Ancient World with our own eyes was a humbling experience. Completed around 2560 B.C., the largest of the Great Pyramids, the Pyramid of Khufu, was for 3800 years the tallest building in the world until the Lincoln Cathedral was built in 1300 A.D. I thought I got a good deal when I bargained for our camels, but later that day my brother just laughed when he heard what I paid. Tracy can ride much better than I. In Saqqara we explored the crumbling remains of the Pyramid of Userkaf, built as the tomb of 5th Dynasty ruler Userkaf and dating to approximately 2458 B.C. While the dressed stone exterior is largely in disrepair and crumbling, the pyramid still rises 163 ft above the surrounding desert. Nearby we also checked out the Step Pyramid of Djoser, built in approximately 2611 B.C. for Djoser (Zoser), 2nd king of the 3rd Dynasty. Rising 204 ft in height, the Step Pyramid of Djoser predated the great pyramids and was the highest building in the world at that time.
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| Great pyramids, visitor rides a camel across the sands to see the pyramids, Egypt. Pyramids of Queens, Pyramid of Menkaure, Pyramid of Khafre, Pyramid of Khufu (left to right, front to back). Giza, Egypt. Image: 00375 Location: Giza, Egypt View this Image in Google Earth! |
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| Pyramid of Userkaf. Saqqara, Egypt. Image: 02574 Location: Saqqara, Egypt View this Image in Google Earth! |
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| Step pyramid of Djoser (Zoser). Saqqara, Egypt. Image: 02575 Location: Saqqara, Egypt View this Image in Google Earth! |
Photo of Egyptian Hieroglyphics
Everywhere we went in Egypt, we saw hieroglyphics. I never got tired of looking at them. The richest area for seeing hieroglyphic carvings was in the necropolis of ancient Thebes, most notably the Luxor and Karnak Temple complexes, Ramesseum and at Medinet Habu. We also saw some hieroglyphic carvings in Saqqara near Cairo, and of course on many of the artifacts at the Egyptian Museum in Cairo. In some places we even saw tourist graffiti from the 1800’s carved alongside the three-millenia-old hieroglyphics.
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| Heiroglyphics. Luxor, Egypt. Image: 02579 Location: Luxor, Egypt |
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| Heiroglyphics and tourist graffiti. Luxor, Egypt. Image: 02580 Location: Luxor, Egypt |
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| Heiroglyphics. Luxor, Egypt. Image: 02595 Location: Luxor, Egypt |
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| Wall detail, Luxor Temple. Luxor, Egypt. Image: 18474 Location: Luxor, Egypt View this Image in Google Earth! |
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| Wall detail with hieroglyphics, Luxor Temple. Luxor, Egypt. Image: 18480 Location: Luxor, Egypt View this Image in Google Earth! |
Photo of the Valley of the Kings
While we were in Luxor, we spent an afternoon visiting the Valley of the Kings, site of the tombs of the New Kingdom pharoahs from the 16th to 11th centuries BC, including the most famous one of all, that of Tutankhamun. The valley itself is quite stark and barren, with little sign above ground of the riches that were once (and perhaps still are) buried there. It was fun descending into some of the tombs and seeing wall paintings that were thousands of years old.
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| Ancient wall paintings,Valley of the Kings. Luxor, Egypt. Image: 18476 Location: Luxor, Egypt View this Image in Google Earth! |
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| Valley of the Kings, roadway leading from Nile River to a complex of ancient tombs. Luxor, Egypt. Image: 18501 Location: Luxor, Egypt View this Image in Google Earth! |
Although the burial mask of Tutankhamun (actually there were several) is now displayed at the Egyptian Museum in Cairo, it was pulled from Tut’s tomb in the Valley of the Kings when it was discovered in 1922.
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| Burial mask of King Tutankhamen, Egyptian Museum. Cairo, Egypt. Image: 18497 Location: Cairo, Egypt View this Image in Google Earth! |
Molokai Reflections
For those winters I worked with Dan Salden and his whale research team in Maui, I stayed with my friend Skip at his surf pad, a quiet condo just yards from the water on Honokeana Cove, in Napili, not far from Kapalua. Most mornings before we drove into town to join Dan and setup the boat, while Skip was brewing up some coffee, listening to the daily weather forecast and phoning his clients, I would walk out to the point and check out Molokai, just across the channel. What a view.
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| Molokai and water pools, viewed from west Maui. Maui, Hawaii, USA. Image: 00253 Location: Maui, Hawaii, USA View this Image in Google Earth! |
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| Molokai summit and cloud, viewed from west Maui. Molokai, Hawaii, USA. Image: 00254 Location: Molokai, Hawaii, USA View this Image in Google Earth! |
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| Clouds and sunlight. Maui, Hawaii, USA. Image: 05643 Location: Maui, Hawaii, USA |
Photo of Karnak Temple
The Karnak Temple complex is part of the necropolis of ancient Thebes, across the Nile River from what is now Luxor, Egypt. Built over a period of 1300 years, Karnak is actually a group of ancient temples and buildings, the assemblage of which is the largest example of a single-location temple complex in the world. The most impressive part of Karnak is the Great Hypostyle Hall, but the entire place is amazing and several days can easily be spent exploring and admiring the ruins. Karnak was the highlight of our entire time in Egypt. Shown below is a section of Karnak, with temple ruins, pylon and a large obelisk visible, as well some of the sphinxes seen along the Avenue of Sphinxes on the approach to the temple of Amun in Karnak.
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| Karnak Temple complex. Luxor, Egypt. Image: 18483 Location: Luxor, Egypt View this Image in Google Earth! |
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| Avenue of Sphinxes approaching the Temple of Amun, part of the Karnak Temple complex. Luxor, Egypt. Image: 18485 Location: Luxor, Egypt View this Image in Google Earth! |
Photo of the West Maui Mountains
We spent a good part of seven winters working with Dan Salden, studying humpback whales (Megaptera novaeangliae) in the basin and channels between Maui, Molokai, Lanai and Kahoolawe islands, occasionally visiting the weather side of Lanai or the rugged, waterfall-laden backside of Molokai. Each morning we would set out from Lahaina harbor and motor as far as the wind and our sunscreen allowed, gradually returning to Lahaina throughout the day as the trades clocked around the West Maui mountains and across filled in on the Auau channel, creating a windline to force us back. A typical day would have us on the water from about 9am until 3pm or so, most of it in the lee of the West Maui mountains. I never grew tired of seeing those mountains, the remnants of a long extinct volcano, rise above Lahaina and the communities of Kaanapali, Kahana, Napili and Kapalua, the cane fields at Olowalu and above Honolua Bay.
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| West Maui mountains rise above the coast of Maui, with clouds flanking the ancient eroded remnants of a volcano. Maui, Hawaii, USA. Image: 05861 Location: Maui, Hawaii, USA |
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| West Maui mountains rise above the coast of Maui, with clouds flanking the ancient eroded remnants of a volcano. Maui, Hawaii, USA. Image: 05863 Location: Maui, Hawaii, USA |
Underwater Photos of Roatan, Honduras
Some years ago I spent two weeks diving on the Honduran island of Roatan with some friends, checking out the corals and walls and just relaxing. The diving was pretty tame, with not much going on underwater. The place was nice for Caribbean corals, with lots of healthy displays of elkhorn and staghorn corals and some big barrel sponges. But the reef fish were totally cleaned out — it was damn near impossible to find an adult grouper, and I don’t think we saw a single shark during the entire two weeks. Unfortunately, grouper was the #1 item on the menu at the place we stayed as well as at all the restaurants we visited, so the paucity of fish life had an obvious explanation. The big attraction at one of the island resorts was a dolphin “encounter”, where people pay to swim in a shallow pen with some captive dolphins. Can you spell w-r-o-n-g in Honduran? It was screwed up in so many ways. It’s too bad, Roatan would be a pleasant place for a dive trip if the fish were still there and the dolphins weren’t caged.
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| Elkhorn coral. Roatan, Honduras. Image: 05563 Species: Acropora palmata Location: Roatan, Honduras View this Image in Google Earth! |
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| Elkhorn coral. Roatan, Honduras. Image: 18502 Species: Acropora palmata Location: Roatan, Honduras View this Image in Google Earth! |
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| Pillar coral. Roatan, Honduras. Image: 18503 Species: Dendrogyra cylindrus Location: Roatan, Honduras View this Image in Google Earth! |
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| Pillar coral. Roatan, Honduras. Image: 18504 Species: Dendrogyra cylindrus Location: Roatan, Honduras View this Image in Google Earth! |
Photo of the Mosque of Ibn Tulun
The Mosque of Ibn Tulun was built in 879 by Ahmad ibn Tulun who, though born the son of a Turkish slave, rose to become the independent governor of Egypt and whose family continued to rule until 905. The mosque bears design elements of his childhood home of Samarra. Unusual are the pointed arches pictured here, which differ from the marble columns which are more typical of Cairene architecture of that period. The arches, laid in offset concentric rows four deep, repeat around the rectangular periphery of the mosque’s enormous courtyard. The photographs were taken prior to the 1999 renovation of the mosque.
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| Arches, Mosque of Ibn Tulun. Cairo, Egypt. Image: 02601 Location: Cairo, Egypt View this Image in Google Earth! |
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| Arches, Mosque of Ibn Tulun. Cairo, Egypt. Image: 02605 Location: Cairo, Egypt View this Image in Google Earth! |
Photo of the Hand of Ozymandias
The classic sonnet “Ozymandias” by Percy Bysshe Shelley has within it two stark ironies. Can you discern them both?
I met a traveller from an antique land
Who said: Two vast and trunkless legs of stone
Stand in the desert. Near them on the sand,
Half sunk, a shatter’d visage lies, whose frown
And wrinkled lip and sneer of cold command
Tell that its sculptor well those passions read
Which yet survive, stamp’d on these lifeless things,
The hand that mock’d them and the heart that fed.
And on the pedestal these words appear:
“My name is Ozymandias, king of kings:
Look on my works, ye mighty, and despair!”
Nothing beside remains: round the decay
Of that colossal wreck, boundless and bare,
The lone and level sands stretch far away. — Percy Bysshe Shelley
While Ozymandias tells the mighty trailing him in history to despair at the unreachable loftiness of his works, in fact all that remains of the arrogant pharaoh is a shattered statue bereft of majesty. The rulers following him should indeed despair, not through their inability to equal his glory but rather by realizing they too are destined for obscurity, as Ozymandias is, by the ravages of time. A second irony stems from the notion that Ozymandias’ place in history is sealed not by the permanence and grandeur of the monuments that he erected, but by the fact that their decay is extreme enough to have caught the eye of the nameless author, whose simple tale of Ozymandias’ ruin will remain when all physical sign of once powerful king is forever gone.
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| Hands of Ozymandias, Ramesseum. Luxor, Egypt. Image: 02598 Location: Luxor, Egypt View this Image in Google Earth! |
Ozymandius is widely considered to be Ramses II, the 13th century BC pharoah who ruled Egypt for 67 years. The historical mark he left upon his country is unrivaled by any other pharoah, and is measured in the archaelogical record by buildings, temples and statues. One of Ramses’ principal icons, known now as the Ozymandias Colossus, would have towered 50′ were it still standing. As it is, the toppled colossus is broken and in decay, and is thought to have inspired Shelley in the penning of his sonnet. You can find the Hands of Ozymandias for yourself at the Ramesseum, the mortuary temple complex for Ramses II that is part of the sprawling necropolis of ancient Thebes, located across the Nile River from modern-day Luxor, Egypt.
Interestingly, Shelley’s friend Horace Smith wrote a similar sonnet, dealing with the same subject, and published it in the same magazine just a month later. Compare and contrast, if you will, Smith’s version with Shelley’s:
In Egypt’s sandy silence, all alone,
Stands a gigantic Leg, which far off throws
The only shadow that the Desert knows:
“I am great OZYMANDIAS,” saith the stone,
“The King of Kings; this mighty City shows
“The wonders of my hand.” The City’s gone,
Nought but the Leg remaining to disclose
The site of this forgotten Babylon.
We wonder, and some Hunter may express
Wonder like ours, when thro’ the wilderness
Where London stood, holding the Wolf in chace,
He meets some fragments huge, and stops to guess
What powerful but unrecorded race
Once dwelt in that annihilated place. — Horace Smith
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Updated: February 3, 2012




























































