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The Earth's three highest mountains - by three definitions

lpetrich

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"Ee-vrist" as opposed to "Eh-vrest".

1976 Standard Atmosphere Calculator - for the average pressure and temperature at different altitudes. For sea level, it has 1 atm (definition) and 15 C, and at Mt. Everest's peak, it's 0.31 atm and -43 C.

 Mountaineering -  World altitude record (mountaineering)
Mountaineering is sometimes called alpinism.

 Ötzi was found in a glacier in the Ötztal Alps near the Austria - Italy border at an altitude of around 3210 m. He lived around 3350 and 3105 BCE, over 5000 years ago. He was at 0.67 atm and -6 C.

The highest known premodern climb was of Llullaillaco in the Andes in Argentina and Chile at 6739 m. Three sacrificed children were found on its peak, sacrificed by Incas around 1500 CE. That was at 0.42 atm and -29 C.

There are many accounts of mountain climbing over the centuries. "A commonly cited example is the 1492 ascent of Mont Aiguille (2,085 m (6,841 ft)) by Antoine de Ville, a French military officer and lord of Domjulien and Beaupré." Not as much as Ötzi, I must note, and at 0.78 atm and 1.4 C.

"The Age of Enlightenment and the Romantic era marked a change of attitudes towards high mountains. In 1757 Swiss scientist Horace-Bénédict de Saussure made the first of several unsuccessful attempts on Mont Blanc in France. He then offered a reward to anyone who could climb the mountain, which was claimed in 1786 by Jacques Balmat and Michel-Gabriel Paccard. The climb is usually considered an epochal event in the history of mountaineering, a symbolic mark of the birth of the sport."

Mt. Blanc is at 4807.81 m, at 0.55 atm and -16 C.

"One of the most dramatic events was the spectacular first ascent of the Matterhorn in 1865 by a party led by English illustrator Edward Whymper, in which four of the party members fell to their deaths. By this point the sport of mountaineering had largely reached its modern form, with a large body of professional guides, equipment, and methodologies."

Matterhorn "meadow peak", also Cervino and Cervin, is in the Alps in Switzerland and Italy, at 4478 m, at 0.57 atm and -14 C.

Low temperatures and low oxygen are only two of the hazards that climbers face with these very high mountains. Steep, rocky slopes, crevasses in glaciers (big cracks in them), avalanches, bad weather, ...
 

Elixir

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Sounds like Mauna Kea would be the hardest to hike from base to summit.
That's not how it works. Terrain and weather conditions are the biggest factors, not total distance or elevation gain. K2 is not as high as Everest, but is a much more challenging peak, and more people die attempting K2 than Everest.
Sure. But think about attempting Mauna Kea from its *base*!
Hiking under 19,000 feet of water can be a real challenge; the ascent to sea level might not be difficult, but the pressure at the bottom is a bitch. In fact the survival rate is zero AFAIK.
 

lpetrich

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From Wiki,
In the Himalaya yaks have been reported at heights of up to 6,100 m (20,000 ft) and the summer snow line can be as high as 6,500 m (21,300 ft). It is likely that local inhabitants went to such heights in search of game, and possibly higher while exploring trade routes, but they did not live there, and there is no evidence that they attempted to climb the summits of the Himalaya before the arrival of Europeans.[4]

In August 1855, the Bavarian brothers Adolf and Robert Schlagintweit of the Magnetic Survey of India made an attempt to climb Kamet (7,756 m), in the Garhwal region of Uttarakhand, India, near the Tibetan border. Spending 10 days above 17,000 ft (5,200 m) they approached the mountain from the Tibetan side, climbing the northwest ridge of the subsidiary peak Abi Gamin. From their highest camp at 19,325 ft (5,890 m), they and some of their guides and carriers reached an altitude of 22,259 ft (6,785 m) according to their barometric measurements, which would have put them higher than Llullaillaco. [5][6]

Many early claims of world altitude records are muddied by incomplete surveying and lack of knowledge of local geography, which have led to reassessments of many of the heights which were originally claimed. In 1862 a khalasi (an Indian assistant of the GTS) climbed Shilla, a summit in Himachal Pradesh which was claimed to be over 7,000 m (23,000 ft) high. More recent surveys have, however, fixed its height at 6,111 m (20,049 ft).[7] Three years later William Johnson of the GTS claimed to have climbed a 7,284 m (23,898 ft) peak during an illicit journey into China, but the mountain he climbed has since been measured at 6,710 m (22,014 ft).[7]
The article then discussed various other disputed records, then got to more recent ones.
On 14 January 1897, Matthias Zurbriggen went on to make the first recorded ascent of Aconcagua in the Andes. Aconcagua is 6,962 metres (22,841 ft) high and, if the claims of Boss and Graham are discounted, was still the highest point to have been reached at that time.[15]

...
An undisputed new altitude record was achieved in 1909 by the Duke of the Abruzzi's expedition to the Karakoram. After failing to make progress on K2 the Duke led an attempt on Chogolisa, where they reached a height of approximately 7,500 m (24,600 ft) before turning around just 150 m below the summit due to bad weather and the risk of falling through a cornice in poor visibility.[20]

The undisputed summit record, though not the altitude record, was broken by 8 meters on June 14, 1911, when the Scottish chemist, explorer, and mountaineer Alec Kellas together with the Sherpas "Sony" and "Tuny's brother" climbed the 7,128 metres (23,386 ft) high Pauhunri on the border of Sikkim and Tibet. Until the late 20th century this mountain was thought to be only 7,065 metres (23,179 ft), so this record was not realized at the time.[21]
British explorers made several attempts to climb Mt. Everest in the 1920's and 1930's, getting up to 8570 m. Some of the climbers brought oxygen with them. The article noted some summit records between the wars, ending with
The summit record was raised once more before the Second World War brought an effective halt to mountaineering in the Himalaya. Nanda Devi, at 7,816 m (25,643 ft) the highest mountain wholly within the British Empire, had been the object of several expeditions, and it was finally climbed on 29 August 1936 by Bill Tilman and Noel Odell.[29]
So by WWII, the height record was 8570 m and the mountain-peak record was a little less, 7816 m. The pressures and temperatures are 0.32 and -41 C, and 0.36 atm and -36 C.
 

lpetrich

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There was one further improvement on the summit record before Everest was conquered. On 3 June 1950 Annapurna (8,091 m, 26,545 ft) became the first 8,000 m mountain to be climbed when the French climbers Maurice Herzog and Louis Lachenal reached its summit on the 1950 French Annapurna expedition. Both Herzog and Lachenal lost their toes to frostbite; Herzog also lost most of his fingers.[31]

The first attempt to climb Everest from the south was made by a Swiss team in 1952. The expedition's high point was reached by Raymond Lambert and the team's Nepali Indian sardar Tenzing Norgay on 26 May, when they reached a point approximately 200 m (650 ft) below the South Summit before turning around in the knowledge that they would not reach the summit in daylight. Their estimated height of 8,600 m (28,210 ft) was slightly higher than the previous altitude record set by the British on the north side of the mountain.[32] The Swiss made further attempts later in May, and again in autumn after the monsoon, but did not regain Lambert and Tenzing's high point.

Mount Everest was climbed the following year. On 26 May, three days before the successful attempt, Tom Bourdillon and Charles Evans reached the South Summit before turning back due to malfunctioning oxygen apparatus. Their height of 8,760 m (28,750 ft) represented a new, short lived, altitude record, and can be seen as a summit record if this is taken to include minor tops as well as genuine mountains.[33] Edmund Hillary and Tenzing Norgay finally reached the 8,848 m (29,029 ft) true summit on 29 May 1953, marking the final chapter in the history of the mountaineering altitude record.[34] While the exact height of Everest's summit is subject to minor variation due to the level of snow cover and the gradual upthrust of the Himalaya, significant changes to the world altitude record are now impossible.
Mt. Everest is about 237 meters higher than K2, the second highest, so revised measurements are unlikely to change its status.
 

lpetrich

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Llullaillaco is one of several Andean mountains with evidence of pre-Columbian climbers:  List of Andean peaks with known pre-Columbian ascents

If you wish to visit Mt. Everest, your first stop is Nepal's capital and largest city,  Kathmandu It has an altitude of 1400 meters, a population of 975 thousand people, and a full-scale airport: Tribhuvan International Airport. The air pressure and temperature are 0.84 atm and 6 C.

A common stop on the way is  Lukla, about 140 km from Kathmandu. It is a small town in the mountains, with a small airport: Tenzing-Hillary Airport.
The airport's paved asphalt runway is accessible only to helicopters and small, fixed-wing, short-takeoff-and-landing aircraft such as the De Havilland Canada DHC-6 Twin Otter, Dornier 228, L-410 Turbolet and Pilatus PC-6 Turbo Porter. The runway is 527 m (1,729 ft) × 30 m (98 ft) with an 11.7% gradient.[1] The airport's elevation is 9,334 ft (2,845 m).[1] The airport is used for passenger flights and for transporting most of the building materials and cargo to Lukla and other towns and villages to the north of Lukla, as there is no road to this region.
The air pressure and temperature are 0.71 atm and -3 C.

That airport is challenging to land on, because it is on a mountain slope, with the mountain rising northeast of the airport.
Due to the difficulties of successfully landing at the airport, the Civil Aviation Authority of Nepal sets high standards, according to which only experienced pilots, who have completed at least 100 short-takeoff-and-landing (STOL) missions, have over one year of STOL experience in Nepal, and have completed ten flights into Lukla with a certified instructor pilot, are allowed to land at the airport.[11][12]
One can walk or go by helicopter the rest of the way to Mt. Everest.

About 13 kilometers north is  Namche Bazaar at 3440 m, with air pressure 0.65 and temperature -7 C. This town has a population of about 1,600, and it is often used as a stop for altitude acclimatization. Mt. Everest is 30 km to the northeast.
 

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Sounds like Mauna Kea would be the hardest to hike from base to summit.
That's not how it works. Terrain and weather conditions are the biggest factors, not total distance or elevation gain. K2 is not as high as Everest, but is a much more challenging peak, and more people die attempting K2 than Everest.
Well, I mean... there's lava to consider... and an ocean...
 

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Having gone the highest, let us now go the lowest.

Relative to sea level, the lowest place is the  Challenger Deep in the  Mariana Trench at 10920 meters, giving a water pressure of a little over 1000 atm.

The lowest dry land is the shores of the Dead Sea, now about 430 m below sea level. Its maximum depth is 298 m, giving a total of 728 m.

The closest solid surface to the Earth's center is the bottom of the  Amundsen Basin at the North Pole. It has a depth of 4500 m below the ocean surface, giving a distance of 6352 km from the Earth's surface. By comparison, the lowest point of the Mariana Trench is at 6366 km from the Earth's center.

Looking at the lowest bits of dry land, they are the  Gould Coast at the southern end of the Ross Sea and the  Most northerly point of land (some island north of Greenland). If one accepts sea ice as a dry surface, then the North Pole is the lowest of all, at 6357 km from the Earth's center, with the Gould Coast being 196 m farther out than it, and that north-of-Greenland island being 257 m farther out.
 

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In his excellent narrative Into Thin Air, Jon Krakauer describes some of the difficulties in ascending and then descending Mt. Everest's peak.
Jon Krakauer said:
Straddling the top of the world, one foot in Tibet and the other in Nepal, I cleared the ice from my oxygen mask, hunched a shoulder against the wind, and stared absently at the vast sweep of earth below. I understood on some dim, detached level that it was a spectacular sight. I'd been fantasizing about this moment, and the release of emotion that would accompany it, for many months.

But now that I was finally here, standing on the summit of Mount Everest, I just couldn't summon the energy to care.
It was the afternoon of May 10. I hadn't slept in 57 hours. The only food I'd been able to force down over the preceding three days was a bowl of Ramen soup and a handful of peanut M&M's. Weeks of violent coughing had left me with two separated ribs, making it excruciatingly painful to breathe.

Twenty-nine thousand twenty-eight feet up in the troposphere, there was so little oxygen reaching my brain that my mental capacity was that of a slow child. Under the circumstances, I was incapable of feeling much of anything except cold and tired. ... I ... started down....

All told, I'd spent less than five minutes on the roof of the world. After a few steps, I paused to take another photo, this one looking down the Southeast Ridge, the route we had ascended. Training my lens on a pair of climbers approaching the summit, I saw something that until that moment had escaped my attention. To the south, where the sky had been perfectly clear just an hour earlier, a blanket of clouds now hid Pumori, Ama Dablam, and the other lesser peaks surrounding Everest.

Days later -- after six bodies had been found, after a search for two others had been abandoned, after surgeons had amputated the gangrenous right hand of my teammate Beck Weathers -- people would ask why, if the weather had begun to deteriorate, had climbers on the upper mountain not heeded the signs? Why did veteran Himalayan guides keep moving upward, leading a gaggle of amateurs, each of whom had paid as much as $65,000 to be ushered safely up Everest, into an apparent death trap? Nobody can speak for the leaders of the two guided groups involved, for both men are now dead. But I can attest that nothing I saw early on the afternoon of May 10 suggested that a murderous storm was about to bear down on us.

To my oxygen-depleted mind, the clouds drifting up the grand valley of ice known as the Western Cwm looked innocuous, wispy, insubstantial.
...
As I began my descent, I was indeed anxious, but my concern had little to do with the weather. A check of the gauge on my oxygen tank had revealed that it was almost empty. I needed to get down, fast.
 

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Now for the lowest relative to its surroundings.

Looking at caves, I find that  Krubera Cave is the deepest known one, with a depth of 2197 m. It is in Asian Georgia near the northeast coast of the Black Sea, with its nearby land surface about 2256 m above sea level. Its lowest point is 59 m above sea level and its distance from the Earth's center is 6368 km.

Looking at canyons, there is a problem. Some canyons are surrounded by very high mountains, like Yarlung Tsangpo Grand Canyon and Kali Gandaki Gorge, both in the Himalaya Mountains in Nepal. So I looked for canyons in plateaus, like the Grand Canyon in the southwestern contiguous US. Its depth is around 1600 meters and its lowest points about 800 m above sea level and about 6371 km from the Earth's center.

In  Canyon I found mention of the  Cotahuasi Canyon and  Colca Canyon of Peru.

Cotahuasi: "With a depth of approximately 3,354 metres (11,004 ft), it is more than twice as deep as the Grand Canyon. The Cotahuasi River eroded the canyon between two mountain massifs: the Coropuna (6,425 m or 21,079 ft ASL) and the Solimana (6,093 m or 19,990 ft ASL)."

Colca: "With a depth of about 1000 - 2000 m (3300 - 6600 ft) (whereas bottom is at cca 2000 m and edges are at 3000 - 4000 metres above the sea level), [1] it is one of the deepest canyons in the world."

But the surrounding topogaphy of both of them is rather mountainous, and not very obviously flat like that around the Grand Canyon.

Turning to the Mariana Trench again, and found ocean-depth charts in  Bathymetry As best as I could read off, that trench's surroundings have depths of 4500 - 5500 meters, so I'll use 5000 m. That makes that trench's lowest point some 5000 m below its surroundings, beating the Grand Canyon and the two Peruvian canyons.
 

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Now the history of trying to reach these depths.  History of underwater diving -  Timeline of diving technology

The first kind of diving was what we now call freediving. That's diving while holding one's breath. That has been done for centuries to collect sea sponges and bivalves, to salvage valuables from sunken ships, and military missions. Bivalves would often be collected to search for pearls that they had made around stray sand grains and the like. This involved collecting a *lot* of these shellfish, thus making pearls very rare. Nowadays, pearls are farmed by giving some oysters or mussels some little balls for them to make mineral deposits around. That aside, one can go down about 30 meters.

Diving bells have been used for centuries. "In 1658, Albrecht von Treileben was contracted by King Gustavus Adolphus of Sweden to salvage the warship Vasa, which sank outside Stockholm harbor in about 32 metres (105 ft) of water on its maiden voyage in 1628. Between 1663 and 1665 von Treileben's divers were successful in raising most of the cannon, working from a diving bell with an estimated free air capacity of about 530 litres (120 imp gal; 140 US gal) for periods of about 15 minutes at a time in dark water with a temperature of about 4 °C (39 °F).[16][17]"

Diving suits date back to around then, often supplied with air from the surface in hoses, though air tanks were developed in the late 18th cy. By the early 19th cy., "standard diving dress" has reached its familiar form, a spherical metal "diving helmet" with windows and waterproof fabric for the rest of the diver's body. Submarines were developed over this time, with the first use of one in warfare was  Turtle (submersible) in 1776 by North American rebels against British ships. A "submersible" is a small submarine, especially one connected to a surface ship with cables.

In the middle of the 19th cy., "decompression sickness" or "the bends" became recognized. it was first recognized in caisson workers, caissons being pressurized cylinders placed on riverbeds to permit the construction of bridge piers and dam foundations. Their pressurization is to keep water from seeping in, and the caisson workers have to be gotten up to working pressure and later gotten back down to normal atmospheric pressure. In the late 19th cy., one doctor recommended a pressure limit of 4 atm, for a depth of 40 meters.
 
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