The following is an article from the science humor magazine Annals of Improbable Research.
Note how the bottom of the Grand Canyon (in this rare view) is darker than at the top, giving an idea of how solar radiation attenuates the deeper one goes into the canyon. (Image credit: Wikipedia user chensiyuan)
By Earle E. Spamer
American Philosophical Society, Philadelphia, Pennsylvania
For a long time now, people visiting the Grand Canyon of the Colorado River have been told it is about 12 to 18 miles wide and “more than” a mile deep, convinced that its majestic views are due to the incessant powers of erosion. New analyses now indicate that the canyon is far larger than imagined. It can never be accurately measured and may unexpectedly hold the answers to long-held geological and environmental mysteries.
Fooling Some of the People
In 1903, President Theodore Roosevelt was the first to recognize generally that there was a problem with the Grand Canyon. Speaking near the canyon rim, he urged Americans to “keep it as it is.” He foresaw an endangered canyon, one that without our intervention would deliver a dwindled heritage to the “children’s children” of his audience. His plea, though widely repeated, went unheeded for decades.
Twenty years ago I devised a plan to place the Grand Canyon in stasis, until such time that the problem of erosion can be remedied. The Final Report1 called for filling the canyon with styrofoam packing material, or “piffles”—some 291 quadrillion of them, to be precise. This plan was also promoted by Grand Canyon National Park.2 I was interviewed a number of times for newspapers and radio to explain myself, and Internet features have taken up the cause.3 A few years later, my Further Final Report4 revealed that the plan had failed because, by mistake, biodegradable piffles had been used—at least, that was the conclusion. This was not correct.
The follow-up report went on to recommend that to safeguard an unprotected canyon, we should ensure that it not be visited at all, nor written about, nor photographed. I argued that all available superlatives have been overextended, and that the number of photons removed from the national park in tourist’s cameras have reached alarming numbers, greatly reducing the amount available for appreciation by our own children’s children. It would be pointless for most of us to visit the canyon anyway. It has been pre-appreciated for us, professionally, by celebrated writers, poets, singers, painters, photographers and filmmakers; our amateur efforts would be lacklustrously insufficient, depreciating further the superlatives and photons.
Still later I postulated that the Grand Canyon may be a fake.5 Had we been all along deceived about the identity or location of the canyon? Hundreds of other places around the world— and on other worlds—claim to be a (or the) Grand Canyon.6 I charged readers to campaign, vigorously, to have the “one true Grand Canyon” designated as the Official Grand Canyon. Apparently, our representatives in Congress and the editors of the world’s great newspapers have thought otherwise, ignoring the pleas of constituents and readers.
Straight Down to China
Now there is conclusive evidence to show that the Grand Canyon is not a fake. Rather, we have deluded ourselves into believing that it is disappearing due to erosion, when in fact it is far more immense than supposed. We have accumulated the evidence for a century and more, but have failed to recognize its significance.
John Hance was one of the early Grand Canyon pioneers, pecking out asbestos from a few mines, then doing much better business in the lucrative tourist trade. By his own admission, he dug the canyon. Hance said he once tried to take his horse across the chasm. “He give a fine big jump—but when we was ‘bout halfway over I seed we couldn’t make it, so I turned him back.”7 This gives us some inkling about how vast the canyon was even a century ago. Another Grand Canyon pioneer–miner, Louis Boucher, reported that the side canyon where he lived “is so narrow and deep that [I see] stars from its recesses at midday.”8 Some years earlier, a traveler reported that elsewhere in the canyon, “So deep and narrow became the rift that the darkness of twilight was experienced, and stars appeared in the narrow line of sky.”9 Yet another tourist corroborated that “From the bottom of the canyon you can look up and see the stars at noonday as plain as though it were night time.”10
Lest one find so many credible reports incredible, first consider that at least one geography text teaches that the canyon contains “tributary gorges, into whose depths the sun penetrates scarcely two hours a day.”11 The Grand Canyon is so deep that even solar radiation attenuates dramatically there, rarely reaching the bottom.
Further substantiating these reports and anlayses is a startling report of “bottomless but colorful abysses of the Grand Canyon.”12 Seeing colors may contradict the attenuation of light, but we may presume, too, that the traveler had a flashlight.
John Hance himself likewise declared that he “coun’t see fer into th’ canyon.” If you dismiss Hance as a yarn-spinner, consider the testimony of Chester T. Crowell, one of the many professional authors who have pre-appreciated the Grand Canyon for us. About his first look at the canyon, he said, “If I had seen a Chinaman peeking at me thru the bottom it would not have surprized me. And then they told me that I could see only half way down from the top…”13 One may just as likely blame the liberal application of whiskey to parched palates on the American frontier for many of the first-hand reports cited here. Then again, a well-loosened tongue can offer great and uninhibited truths.
What Can Be Seen at the Grand Canyon?
We have clear, empirical observations of the attenuation of views of the Grand Canyon. How, then, may we account for the billions of mostly amateur photographs taken of the Grand Canyon that display sweeping, majestic panoramas, clear to the bottom of the canyon? I have not slept in years thinking about this. But modern computers save the day.
An advertisement for Sales Cycle Manager software from Sales Ways promises “More Views Than the Grand Canyon!”14 This begs the question, so far as I know never before asked, “How many views are there at the Grand Canyon?” Obviously, Sales Ways’s software engineers have derived the answer, since their product offers more views than that. It seems that the precise number is proprietary information.
A simple calculation can be made. If one presumes that every occupiable spot on the rim of the Grand Canyon equals 1.0 View, then if we know the lengths of the rims we can derive the total. There is a problem: Just what constitutes the rim of the Grand Canyon? Precisely defined, it is an event horizon, where gravity has its greatest effect. There, one more step will be, dramatically, one’s last.15 Moreover, the Grand Canyon is so massive that it has two event horizons, or rims, one on each side.
In 1903, two hikers search futilely for the other side the Grand Canyon, dangerously close to the event horizon.
Wegener and the Rim Cycle: A Fractal Fairy Tale
In 1894, the distance around the Grand Canyon was reported to be 450 miles.16 This seems to be a reasonable estimate, but it differs dramatically from a precise measurement made a century later by top government scientists. They measured 2,757 miles.17
So why the great difference between 1894 and 1999? One might suspect the influence of continental drift, first postulated by Alfred Wegener. However, no geodetic measurements have indicated a change in the distance between the two sides of Grand Canyon during the last century. Still it is worth noting that in 1890 the journal Science reported contrarily, “The walls of the Grand Canyon of the Colorado are moving toward each other” and they may eventually touch!18 This is a conundrum that I choose to ignore.19
Now it is becoming clear that what should be very accurate measurements—the rim, depth, and breadth of the Grand Canyon—has everything to do with where and how the canyon is measured. For a long time we have intuitively known the canyon is big—in fact, before it was named the Grand Canyon in the 1860s it was called Big Canyon, a tantalizing clue that early explorers recognized that the canyon was becoming larger the more they looked at it. Yet in all this time we failed to realize that it is even bigger yet.
Benoit B. Mandelbrot’s groundbreaking 1967 paper, “How Long is the Coast of Britain?”20 introduced many scientist to the theory and applications of fractals. Mandelbrot demonstrated that the island’s coast, when not measured with “ordinary curves,” may be instead measured with any number of straight-line segments. As segment vectors and the number of them are changed, astonishingly different results are obtained at ever-finer scales of resolution.
Mandelbrot’s study was tidy, two-dimensional in perspective. Had he sauntered the edge of the Grand Canyon instead of an odyssey around Britain, he may have found himself postulating from a perspective that was three-dimensional, and far deeper. Had he done that, who knows where the science and mathematics of fractals would be today!
Extrapolating the U.S. Geological Survey’s inferentially reliable data on the mileages of the Grand Canyon rims, the formula for total Grand Canyon views is N = 2,757x, where x is the number of vantage points per mile. Fractal science, however, dictates that we may derive up to an infinity of segments between points, thus also an infinity of vantage points. A convincing application is found in Karl-Heinz Becker and Michael Dörfer’s Dynamical Systems and Fractals, where they feature the Grand Canyon throughout the chapter, “Journey to the Land of Infinite Structures.”21 And a sample of actual field images of elusive fractals in their natural canyon habitat are in Gayla Chandler’s online exhibit, “Natural Fractals in Grand Canyon National Park.”22
Science On the Edge: Wonders Beyond Sight
In fractal illustrations, the pointy parts in the diagrams occupy far less space than do edges or planes. The Grand Canyon expresses its fractal nature in having more edge than points. There can be an infinity of viewpoints along the Grand Canyon’s rims, so it should make no difference where one views the canyon. Yet as we observe empirically, people crowd the viewpoints; longer, less interesting edges are infrequently visited. This must be because the viewpoints offer proper view and sun angles that occasionally reveal breathtaking panoramas.
Given that there is much more beyond the view, as fractals intimate, do people honestly see what they go to see? Chester Crowell’s report, cited earlier, retells the account of a man bragging about the places he had seen. “Have you ever had delerium tremens?” asked a second man. “No,” was the reply. “Then you ain’t never seen nothing.”
There are documented sightings of nothing in the Grand Canyon, as previously discussed. One writer further clarified that in the canyon “there ain’t no possible way of paintin’ a sign you could make out with the Lick telescope [and] a full-grown World’s Fair goin’ full blast with the blower on you wouldn’t even see across that Canyon.”23 Hundreds more writers further testify that the canyon is not only indescribable, but unfathomable. The one and truly Grand Canyon must be larger—and may hold more—than we have ever imagined.
Enduring Mysteries—Solved!
Detail of fractal in the Grand Canyon. (Image credit: Flickr user Marcin Wichary)
If the edges of the Grand Canyon can, upon close examination, approach infinite lengths due to ever-increasing crenulation, it stands to reason that the volumetric potential of the canyon must increase likewise, with profound effects. The hypothesis is presented here that the whole of Grand Canyon’s three-dimensional fabric, bound within its two event horizons, must, like the rims, become more crenulated the more precisely it is measured.
If a fractally defined Grand Canyon is a large photonic sink, as is demonstrated by the attenuation of solar radiation in its depths, it must also be a sedimentary sink. For a century and more geologists have searched the Colorado River region downstream from the Grand Canyon for the largely unaccounted-for sediment eroded from the canyon.24 They have followed standard, Newtonian processes of erosion, presuming also that the actions of the river flowing within the canyon carried away the eroded material. They have been searching too far afield.
A fractally derived model of the Grand Canyon suggests that the sediment can still be within a canyon that comprises a densely packed fractal dimension of undetermined complexity, up to an infinity of fractal crenulations. Paradoxically, the Grand Canyon may store within highly crenulated walls and slopes, at scales too small to measure, the very sediments already eroded from them. Likewise, the utter failure to fill the canyon with styrofoam piffles, and the subsequent fruitless search for those missing piffles, provide further evidence for an exceedingly large number of fractal crenulations throughout the Grand Canyon. The attenuation both of canyon views and solar radiation (discussed earlier) may further assist by concealing evidence of the canyon’s intricate fractal sheath.
In the manner of a prediction, as yet untested, an infinitely long plumb line dropped into the canyon will become entangled in a fractal matrix, with no way of determining whether it has struck true bottom. If proved, it may be reasoned that all other measurements of the Grand Canyon can never be precise. Nonetheless, should these and other such steps be taken it will be quite a giant leap for mankind.*
*Note
With apologies to Neil Armstrong. (Both Armstrong and Benoit Mandelbrot are elected members of the American Philosophical Society.25 Although I do not know whether the two men have ever met, both are resolved to exploring the infinite and are, in the words of Society founder Benjamin Franklin and the Society’s mission, “For Promoting Useful Knowledge.”)
Regular Notes
1“Preserving the Grand Canyon: Final report.” in Sex as a Heap of Malfunctioning Rubble (and Further Improbabilities), Marc Abrahams, ed., Workman Publishing, New York, 1993.
2 “Preserving the Grand Canyon; final report,” Nature Notes (Grand Canyon National Park), vol. 12, no. 1, spring 1996, pp. 1–3.
3 For example: David George Gordon, “Filling the Grand Canyon,” Mad or Rad? (ABCnews.com Internet column), 1997.
4 “The Grand Canyon—Further Final Report, and Users’ Guide,” Annals of Improbable Research, vol. 3, no. 4, July–August 1997, pp. 15–8.
5 “Is the Grand Canyon a Fake?”, Annals of Improbable Research, vol. 12, no. 2, March–April 2006, pp. 18–22.
6 “Other Grand Canyons: Incarnations Around the World and on Other Worlds,” Annals of Improbable Research, vol. 12, no. 2, March–April 2006, pp. 24–5. This list is so long that it had to be printed in exceedingly small type. Magnifying glass not provided.
7 As recounted in “Pioneers of the Canyon,” The Mentor 3, October 1, 1915, separate color plate. Also see Don Lago, “Tall Cliffs and Tall Tales: The Origins of John Hance,” The Ol’ Pioneer, vol. 21, no. 3, summer 2010, pp. 1–2 and 5–12.
8 George Wharton James, In and Around the Grand Canyon, Little, Brown, and Co., Boston, 1900.
9 J. G. Lemmon, “Grand Canyon of the Colorado,” The Overland Monthly, Ser. 2, 12, September 1888, p. 253–4.
10 Joe Mitchell Chapple, The Happy Habit, Chapple, Boston, 1908.
11 R. H. Whitbeck, High School Geography, Macmillan, New York, 1922.
12 Elbert Hubbard and George Wharton James, So Here Then Is a Little Journey to the Home of Joaquin Miller, Roycrofters, East Aurora NY, 1903.
13 Chester T. Crowell, “Straight Down to China,” The Independent, June 4, 1921, p. 579. Not to spoil a great title, but the Indian Ocean is actually straight down from the Grand Canyon. And come to think of it, since no one else has, straight down from China is southern South America and the South Atlantic and Pacific Oceans. You read it here first.
14 Sales Cycle Manager, Lotus Notes Edition promotional brochure, Sales Ways, Mississauga, ON, seen in 2010. The use of product names here does not imply an endorsement by me or the publisher of AIR. Sales Ways’s brochure was just a serendipitous find since I have nothing to sell.
15 Michael P. Ghiglieri and Thomas M. Myers, Over the Edge: Death in Grand Canyon; Gripping Accounts of All Known Fatal Mishaps in the Most Famous of the World’s Seven Natural Wonders, Puma Press, Flagstaff, 2001.
16 “A Hole in the Ground Four Hundred and Fifty Miles Around,” Texas Medical Journal, vol. 9, no. 12, June 1894, p. 660.
17 George H. Billingsley and Haydee M. Hampton, “Physiographic Rim of the Grand Canyon, Arizona,” U.S. Geological Survey Open-File Report 99-30, 1999. Although the Grand Canyon was first mapped in 1858, and despite the construction of a few retaining walls and railings on the brink of the canyon by the National Park Service early in the 20th century, it was not until this map was published that the U.S. Geological Survey finally located the entire rim of the Grand Canyon. In their explanatory notes the geologists did not disclose the lengths of the lines. Fortunately, one of the geologists is a good acquaintance of mine. Despite this I was able to get the answer through the Freedom of Information Act. Using ArcInfo, the computer-based data points for the rim lines on the north side of the canyon, following all the wiggles of the lines as mapped, shows the north rim is 1,384 miles long. The south side, even though the waggles are completely different, has a rim 1,373 miles long. This reveals that even though the two rims look significantly different, they are obviously nearly the same, although there is a subtle asymmetry that may have deep significance.
18 Gilbert Thompson, “An Hypothesis for the So-Called Encroachments of the Sea Upon the Land,” Science, vol. 15, May 30, 1890, p. 333.
19 By choosing to ignore this I rely on the “Theriot Effect,” first recognized by David M. Hillis in “Taxonomic Sampling, Phylogenetic Accuracy, and Investigator Bias,” Systematic Biology, vol. 47, no. 1, May 1998, pp. 3–8. This is an effect of a widely used data-sampling technique, named for the landmark study by Edward C. Theriot, Arthur E. Bogan, and Earle E. Spamer, “The Taxonomy of Barney: Evidence of Convergence in hominid Evolution,” Annals of Improbable Research, vol. 1, no. 1, January–February 1995, pp. 3–7 (and since republished worldwide in numerous translations). The effect results from selecting data points until expected results are achieved, then stopping.
20 B. B. Mandelbrot, “How Long Is the Coast of Britain?” Science, vol. 156, 1967, p. 636–8.
21Karl-Heinz Becker and Michael Dörfler, Dynamical Systems and Fractals: Computer Graphics Experiments in Pascal, Cambridge University Press, Cambridge and New York, 1989.
22 Gayla Chandler, “Natural Fractals in Grand Canyon National Park,” http://local.wasp.uwa.edu.au/~pbourke/fractals/grandcanyon/ (accessed in 2010).
23 California John quoted in The Cabin, Stewart Edward White, Doubleday, Page and Co., Garden City, 1911.
24 Richard A. Young and Earle E. Spamer, eds., Colorado River: Origin and Evolution, Grand Canyon Association Monograph 12, 2001.
25 Behold! A footnote within a footnote: http://www.amphilsoc.org
Visit their website for more research that makes people LAUGH and then THINK.
Note how the bottom of the Grand Canyon (in this rare view) is darker than at the top, giving an idea of how solar radiation attenuates the deeper one goes into the canyon. (Image credit: Wikipedia user chensiyuan)
By Earle E. Spamer
American Philosophical Society, Philadelphia, Pennsylvania
For a long time now, people visiting the Grand Canyon of the Colorado River have been told it is about 12 to 18 miles wide and “more than” a mile deep, convinced that its majestic views are due to the incessant powers of erosion. New analyses now indicate that the canyon is far larger than imagined. It can never be accurately measured and may unexpectedly hold the answers to long-held geological and environmental mysteries.
Fooling Some of the People
In 1903, President Theodore Roosevelt was the first to recognize generally that there was a problem with the Grand Canyon. Speaking near the canyon rim, he urged Americans to “keep it as it is.” He foresaw an endangered canyon, one that without our intervention would deliver a dwindled heritage to the “children’s children” of his audience. His plea, though widely repeated, went unheeded for decades.
Twenty years ago I devised a plan to place the Grand Canyon in stasis, until such time that the problem of erosion can be remedied. The Final Report1 called for filling the canyon with styrofoam packing material, or “piffles”—some 291 quadrillion of them, to be precise. This plan was also promoted by Grand Canyon National Park.2 I was interviewed a number of times for newspapers and radio to explain myself, and Internet features have taken up the cause.3 A few years later, my Further Final Report4 revealed that the plan had failed because, by mistake, biodegradable piffles had been used—at least, that was the conclusion. This was not correct.
The follow-up report went on to recommend that to safeguard an unprotected canyon, we should ensure that it not be visited at all, nor written about, nor photographed. I argued that all available superlatives have been overextended, and that the number of photons removed from the national park in tourist’s cameras have reached alarming numbers, greatly reducing the amount available for appreciation by our own children’s children. It would be pointless for most of us to visit the canyon anyway. It has been pre-appreciated for us, professionally, by celebrated writers, poets, singers, painters, photographers and filmmakers; our amateur efforts would be lacklustrously insufficient, depreciating further the superlatives and photons.
Still later I postulated that the Grand Canyon may be a fake.5 Had we been all along deceived about the identity or location of the canyon? Hundreds of other places around the world— and on other worlds—claim to be a (or the) Grand Canyon.6 I charged readers to campaign, vigorously, to have the “one true Grand Canyon” designated as the Official Grand Canyon. Apparently, our representatives in Congress and the editors of the world’s great newspapers have thought otherwise, ignoring the pleas of constituents and readers.
Straight Down to China
Now there is conclusive evidence to show that the Grand Canyon is not a fake. Rather, we have deluded ourselves into believing that it is disappearing due to erosion, when in fact it is far more immense than supposed. We have accumulated the evidence for a century and more, but have failed to recognize its significance.
John Hance was one of the early Grand Canyon pioneers, pecking out asbestos from a few mines, then doing much better business in the lucrative tourist trade. By his own admission, he dug the canyon. Hance said he once tried to take his horse across the chasm. “He give a fine big jump—but when we was ‘bout halfway over I seed we couldn’t make it, so I turned him back.”7 This gives us some inkling about how vast the canyon was even a century ago. Another Grand Canyon pioneer–miner, Louis Boucher, reported that the side canyon where he lived “is so narrow and deep that [I see] stars from its recesses at midday.”8 Some years earlier, a traveler reported that elsewhere in the canyon, “So deep and narrow became the rift that the darkness of twilight was experienced, and stars appeared in the narrow line of sky.”9 Yet another tourist corroborated that “From the bottom of the canyon you can look up and see the stars at noonday as plain as though it were night time.”10
Lest one find so many credible reports incredible, first consider that at least one geography text teaches that the canyon contains “tributary gorges, into whose depths the sun penetrates scarcely two hours a day.”11 The Grand Canyon is so deep that even solar radiation attenuates dramatically there, rarely reaching the bottom.
Further substantiating these reports and anlayses is a startling report of “bottomless but colorful abysses of the Grand Canyon.”12 Seeing colors may contradict the attenuation of light, but we may presume, too, that the traveler had a flashlight.
John Hance himself likewise declared that he “coun’t see fer into th’ canyon.” If you dismiss Hance as a yarn-spinner, consider the testimony of Chester T. Crowell, one of the many professional authors who have pre-appreciated the Grand Canyon for us. About his first look at the canyon, he said, “If I had seen a Chinaman peeking at me thru the bottom it would not have surprized me. And then they told me that I could see only half way down from the top…”13 One may just as likely blame the liberal application of whiskey to parched palates on the American frontier for many of the first-hand reports cited here. Then again, a well-loosened tongue can offer great and uninhibited truths.
What Can Be Seen at the Grand Canyon?
We have clear, empirical observations of the attenuation of views of the Grand Canyon. How, then, may we account for the billions of mostly amateur photographs taken of the Grand Canyon that display sweeping, majestic panoramas, clear to the bottom of the canyon? I have not slept in years thinking about this. But modern computers save the day.
An advertisement for Sales Cycle Manager software from Sales Ways promises “More Views Than the Grand Canyon!”14 This begs the question, so far as I know never before asked, “How many views are there at the Grand Canyon?” Obviously, Sales Ways’s software engineers have derived the answer, since their product offers more views than that. It seems that the precise number is proprietary information.
A simple calculation can be made. If one presumes that every occupiable spot on the rim of the Grand Canyon equals 1.0 View, then if we know the lengths of the rims we can derive the total. There is a problem: Just what constitutes the rim of the Grand Canyon? Precisely defined, it is an event horizon, where gravity has its greatest effect. There, one more step will be, dramatically, one’s last.15 Moreover, the Grand Canyon is so massive that it has two event horizons, or rims, one on each side.
In 1903, two hikers search futilely for the other side the Grand Canyon, dangerously close to the event horizon.
Wegener and the Rim Cycle: A Fractal Fairy Tale
In 1894, the distance around the Grand Canyon was reported to be 450 miles.16 This seems to be a reasonable estimate, but it differs dramatically from a precise measurement made a century later by top government scientists. They measured 2,757 miles.17
So why the great difference between 1894 and 1999? One might suspect the influence of continental drift, first postulated by Alfred Wegener. However, no geodetic measurements have indicated a change in the distance between the two sides of Grand Canyon during the last century. Still it is worth noting that in 1890 the journal Science reported contrarily, “The walls of the Grand Canyon of the Colorado are moving toward each other” and they may eventually touch!18 This is a conundrum that I choose to ignore.19
Now it is becoming clear that what should be very accurate measurements—the rim, depth, and breadth of the Grand Canyon—has everything to do with where and how the canyon is measured. For a long time we have intuitively known the canyon is big—in fact, before it was named the Grand Canyon in the 1860s it was called Big Canyon, a tantalizing clue that early explorers recognized that the canyon was becoming larger the more they looked at it. Yet in all this time we failed to realize that it is even bigger yet.
Benoit B. Mandelbrot’s groundbreaking 1967 paper, “How Long is the Coast of Britain?”20 introduced many scientist to the theory and applications of fractals. Mandelbrot demonstrated that the island’s coast, when not measured with “ordinary curves,” may be instead measured with any number of straight-line segments. As segment vectors and the number of them are changed, astonishingly different results are obtained at ever-finer scales of resolution.
Mandelbrot’s study was tidy, two-dimensional in perspective. Had he sauntered the edge of the Grand Canyon instead of an odyssey around Britain, he may have found himself postulating from a perspective that was three-dimensional, and far deeper. Had he done that, who knows where the science and mathematics of fractals would be today!
Extrapolating the U.S. Geological Survey’s inferentially reliable data on the mileages of the Grand Canyon rims, the formula for total Grand Canyon views is N = 2,757x, where x is the number of vantage points per mile. Fractal science, however, dictates that we may derive up to an infinity of segments between points, thus also an infinity of vantage points. A convincing application is found in Karl-Heinz Becker and Michael Dörfer’s Dynamical Systems and Fractals, where they feature the Grand Canyon throughout the chapter, “Journey to the Land of Infinite Structures.”21 And a sample of actual field images of elusive fractals in their natural canyon habitat are in Gayla Chandler’s online exhibit, “Natural Fractals in Grand Canyon National Park.”22
Science On the Edge: Wonders Beyond Sight
In fractal illustrations, the pointy parts in the diagrams occupy far less space than do edges or planes. The Grand Canyon expresses its fractal nature in having more edge than points. There can be an infinity of viewpoints along the Grand Canyon’s rims, so it should make no difference where one views the canyon. Yet as we observe empirically, people crowd the viewpoints; longer, less interesting edges are infrequently visited. This must be because the viewpoints offer proper view and sun angles that occasionally reveal breathtaking panoramas.
Given that there is much more beyond the view, as fractals intimate, do people honestly see what they go to see? Chester Crowell’s report, cited earlier, retells the account of a man bragging about the places he had seen. “Have you ever had delerium tremens?” asked a second man. “No,” was the reply. “Then you ain’t never seen nothing.”
There are documented sightings of nothing in the Grand Canyon, as previously discussed. One writer further clarified that in the canyon “there ain’t no possible way of paintin’ a sign you could make out with the Lick telescope [and] a full-grown World’s Fair goin’ full blast with the blower on you wouldn’t even see across that Canyon.”23 Hundreds more writers further testify that the canyon is not only indescribable, but unfathomable. The one and truly Grand Canyon must be larger—and may hold more—than we have ever imagined.
Enduring Mysteries—Solved!
Detail of fractal in the Grand Canyon. (Image credit: Flickr user Marcin Wichary)
If the edges of the Grand Canyon can, upon close examination, approach infinite lengths due to ever-increasing crenulation, it stands to reason that the volumetric potential of the canyon must increase likewise, with profound effects. The hypothesis is presented here that the whole of Grand Canyon’s three-dimensional fabric, bound within its two event horizons, must, like the rims, become more crenulated the more precisely it is measured.
If a fractally defined Grand Canyon is a large photonic sink, as is demonstrated by the attenuation of solar radiation in its depths, it must also be a sedimentary sink. For a century and more geologists have searched the Colorado River region downstream from the Grand Canyon for the largely unaccounted-for sediment eroded from the canyon.24 They have followed standard, Newtonian processes of erosion, presuming also that the actions of the river flowing within the canyon carried away the eroded material. They have been searching too far afield.
A fractally derived model of the Grand Canyon suggests that the sediment can still be within a canyon that comprises a densely packed fractal dimension of undetermined complexity, up to an infinity of fractal crenulations. Paradoxically, the Grand Canyon may store within highly crenulated walls and slopes, at scales too small to measure, the very sediments already eroded from them. Likewise, the utter failure to fill the canyon with styrofoam piffles, and the subsequent fruitless search for those missing piffles, provide further evidence for an exceedingly large number of fractal crenulations throughout the Grand Canyon. The attenuation both of canyon views and solar radiation (discussed earlier) may further assist by concealing evidence of the canyon’s intricate fractal sheath.
In the manner of a prediction, as yet untested, an infinitely long plumb line dropped into the canyon will become entangled in a fractal matrix, with no way of determining whether it has struck true bottom. If proved, it may be reasoned that all other measurements of the Grand Canyon can never be precise. Nonetheless, should these and other such steps be taken it will be quite a giant leap for mankind.*
*Note
With apologies to Neil Armstrong. (Both Armstrong and Benoit Mandelbrot are elected members of the American Philosophical Society.25 Although I do not know whether the two men have ever met, both are resolved to exploring the infinite and are, in the words of Society founder Benjamin Franklin and the Society’s mission, “For Promoting Useful Knowledge.”)
Regular Notes
1“Preserving the Grand Canyon: Final report.” in Sex as a Heap of Malfunctioning Rubble (and Further Improbabilities), Marc Abrahams, ed., Workman Publishing, New York, 1993.
2 “Preserving the Grand Canyon; final report,” Nature Notes (Grand Canyon National Park), vol. 12, no. 1, spring 1996, pp. 1–3.
3 For example: David George Gordon, “Filling the Grand Canyon,” Mad or Rad? (ABCnews.com Internet column), 1997.
4 “The Grand Canyon—Further Final Report, and Users’ Guide,” Annals of Improbable Research, vol. 3, no. 4, July–August 1997, pp. 15–8.
5 “Is the Grand Canyon a Fake?”, Annals of Improbable Research, vol. 12, no. 2, March–April 2006, pp. 18–22.
6 “Other Grand Canyons: Incarnations Around the World and on Other Worlds,” Annals of Improbable Research, vol. 12, no. 2, March–April 2006, pp. 24–5. This list is so long that it had to be printed in exceedingly small type. Magnifying glass not provided.
7 As recounted in “Pioneers of the Canyon,” The Mentor 3, October 1, 1915, separate color plate. Also see Don Lago, “Tall Cliffs and Tall Tales: The Origins of John Hance,” The Ol’ Pioneer, vol. 21, no. 3, summer 2010, pp. 1–2 and 5–12.
8 George Wharton James, In and Around the Grand Canyon, Little, Brown, and Co., Boston, 1900.
9 J. G. Lemmon, “Grand Canyon of the Colorado,” The Overland Monthly, Ser. 2, 12, September 1888, p. 253–4.
10 Joe Mitchell Chapple, The Happy Habit, Chapple, Boston, 1908.
11 R. H. Whitbeck, High School Geography, Macmillan, New York, 1922.
12 Elbert Hubbard and George Wharton James, So Here Then Is a Little Journey to the Home of Joaquin Miller, Roycrofters, East Aurora NY, 1903.
13 Chester T. Crowell, “Straight Down to China,” The Independent, June 4, 1921, p. 579. Not to spoil a great title, but the Indian Ocean is actually straight down from the Grand Canyon. And come to think of it, since no one else has, straight down from China is southern South America and the South Atlantic and Pacific Oceans. You read it here first.
14 Sales Cycle Manager, Lotus Notes Edition promotional brochure, Sales Ways, Mississauga, ON, seen in 2010. The use of product names here does not imply an endorsement by me or the publisher of AIR. Sales Ways’s brochure was just a serendipitous find since I have nothing to sell.
15 Michael P. Ghiglieri and Thomas M. Myers, Over the Edge: Death in Grand Canyon; Gripping Accounts of All Known Fatal Mishaps in the Most Famous of the World’s Seven Natural Wonders, Puma Press, Flagstaff, 2001.
16 “A Hole in the Ground Four Hundred and Fifty Miles Around,” Texas Medical Journal, vol. 9, no. 12, June 1894, p. 660.
17 George H. Billingsley and Haydee M. Hampton, “Physiographic Rim of the Grand Canyon, Arizona,” U.S. Geological Survey Open-File Report 99-30, 1999. Although the Grand Canyon was first mapped in 1858, and despite the construction of a few retaining walls and railings on the brink of the canyon by the National Park Service early in the 20th century, it was not until this map was published that the U.S. Geological Survey finally located the entire rim of the Grand Canyon. In their explanatory notes the geologists did not disclose the lengths of the lines. Fortunately, one of the geologists is a good acquaintance of mine. Despite this I was able to get the answer through the Freedom of Information Act. Using ArcInfo, the computer-based data points for the rim lines on the north side of the canyon, following all the wiggles of the lines as mapped, shows the north rim is 1,384 miles long. The south side, even though the waggles are completely different, has a rim 1,373 miles long. This reveals that even though the two rims look significantly different, they are obviously nearly the same, although there is a subtle asymmetry that may have deep significance.
18 Gilbert Thompson, “An Hypothesis for the So-Called Encroachments of the Sea Upon the Land,” Science, vol. 15, May 30, 1890, p. 333.
19 By choosing to ignore this I rely on the “Theriot Effect,” first recognized by David M. Hillis in “Taxonomic Sampling, Phylogenetic Accuracy, and Investigator Bias,” Systematic Biology, vol. 47, no. 1, May 1998, pp. 3–8. This is an effect of a widely used data-sampling technique, named for the landmark study by Edward C. Theriot, Arthur E. Bogan, and Earle E. Spamer, “The Taxonomy of Barney: Evidence of Convergence in hominid Evolution,” Annals of Improbable Research, vol. 1, no. 1, January–February 1995, pp. 3–7 (and since republished worldwide in numerous translations). The effect results from selecting data points until expected results are achieved, then stopping.
20 B. B. Mandelbrot, “How Long Is the Coast of Britain?” Science, vol. 156, 1967, p. 636–8.
21Karl-Heinz Becker and Michael Dörfler, Dynamical Systems and Fractals: Computer Graphics Experiments in Pascal, Cambridge University Press, Cambridge and New York, 1989.
22 Gayla Chandler, “Natural Fractals in Grand Canyon National Park,” http://local.wasp.uwa.edu.au/~pbourke/fractals/grandcanyon/ (accessed in 2010).
23 California John quoted in The Cabin, Stewart Edward White, Doubleday, Page and Co., Garden City, 1911.
24 Richard A. Young and Earle E. Spamer, eds., Colorado River: Origin and Evolution, Grand Canyon Association Monograph 12, 2001.
25 Behold! A footnote within a footnote: http://www.amphilsoc.org
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