APOD: Moonrise and Mountain Shadow (2020 Mar 14)

[APOD Robot]

Moonrise and Mountain Shadow

Explanation: What phase of the Moon is 3.14 radians from the Sun? The Full Moon, of course. Even though the Moon might look full for several days, the Moon is truly at its full phase when it is 3.14 radians (aka 180 degrees) from the Sun in ecliptic longitude. That's opposite the Sun in planet Earth's sky. Rising as the Sun set on March 9, only an hour or so after the moment of its full phase, this orange tinted and slightly flattened Moon still looked full. It was photographed opposite the setting Sun from Teide National Park on the Canary Island of Tenerife. Also opposite the setting Sun, seen from near the Teide volcano peak about 3,500 meters above sea level, is the mountain's rising triangular shadow extending into Earth's dense atmosphere. Below the distant ridge line on the left are the white telescope domes of Teide Observatory

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[BobStein-VisiBone]

Astronomy Pi of the Day.

[Boomer12k]

Splendid...

:---[===] *

[orin stepanek]

Nice catch! Daniel Lopez; It is really a nice Photo!

[neufer]

Teide Two Pi

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Coordinates: 28°16′ 23″ N 16°38′ 22″ W

<<Mount Teide (Spanish: El Teide, Pico del Teide, "Teide Peak") is a volcano on Tenerife in the Canary Islands, Spain. Its summit (at 3,718 m) is the highest point in Spain and the highest point above sea level in the islands of the Atlantic.

If measured from the ocean floor, it is at 7,500 m the fourth-highest volcano in the world, and is described by UNESCO and NASA as Earth's third-tallest volcanic structure. Teide's elevation above sea level makes Tenerife the tenth highest island in the world. Teide is an active volcano: its most recent eruption occurred in 1909 from the El Chinyero vent on the northwestern Santiago rift.

Teide was a sacred mountain for the aboriginal Guanches. According to legend, Guayota (the devil) kidnapped Magec (the god of light and the sun) and imprisoned him inside the volcano, plunging the world into darkness. The Guanches asked their supreme god Achamán for clemency, so Achamán fought Guayota, freed Magec from the bowels of the mountain, and plugged the crater with Guayota. It is said that since then, Guayota has remained locked inside Teide. When going on to Teide during an eruption, it was customary for the Guanches to light bonfires to scare Guayota. Guayota is often represented as a black dog, accompanied by his host of demons (Tibicenas).

The Guanches also believed that Teide held up the sky. Many hiding places found in the mountains contain the remains of stone tools and pottery. These have been interpreted as being ritual deposits to counter the influence of evil spirits, like those made by the Berbers of Kabylie. The Guanches believed the mountain to be the place that housed the forces of evil and the most evil figure, Guayota.>>

Tweetie Pi

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The string 3,718 m occurs at Pi position: 2,774:

• 59695362314429524849 3,718 71101457654035902799

The string 7,500 m occurs at Pi position: 7,366:

• 57623361064250639049 7,500 86562710953591946589

The string 16°38′ 22″ W occurs at Pi position: 1,271,373:

• 08448409499242713417 16 38 22 08971001656088517481

The string 28°16′ 23″ N occurs at Pi position: 2,031,240:

• 79262132388923122293 28 16 23 55886733206547590324

[heehaw]

The string 0123456789876543210 occurs infinitely many times in pi.

[neufer]

The string 0123456789876543210 occurs infinitely many times in pi.

More interesting, perhaps, is the fact that
every number occurs infinitely many times in the continued fraction coefficients of pi:

3;7,15,1,292,1,1,1,2,1,3,1,...

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<<In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on. In a finite continued fraction (or terminated continued fraction), the iteration/recursion is terminated after finitely many steps by using an integer in lieu of another continued fraction. In contrast, an infinite continued fraction is an infinite expression. In either case, all integers in the sequence, other than the first, must be positive. The integers a_i are called the coefficients or terms of the continued fraction.>>

Continued fraction coefficients of irrational numbers:

ϕ = [1;1,1,1,1,1,1,1,1,1,1,1,...] (sequence A000012 in the OEIS). The golden ratio, the irrational number that is the "most difficult" to approximate rationally. See: A property of the golden ratio φ.

√19 = [4;2,1,3,1,2,8,2,1,3,1,2,8,...] (sequence A010124 in the OEIS). The pattern repeats indefinitely with a period of 6.

e = [2;1,2,1,1,4,1,1,6,1,1,8,...] (sequence A003417 in the OEIS). The pattern repeats indefinitely with a period of 3 except that 2 is added to one of the terms in each cycle.

π = [3;7,15,1,292,1,1,1,2,1,3,1,...] (sequence A001203 in the OEIS). No pattern has ever been found in this representation.

Simple continued fraction expansion of Pi:

3, 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, 1, 2, 2, 2, 2, 1, 84, 2, 1, 1, 15, 3, 13, 1, 4, 2, 6, 6, 99, 1, 2, 2, 6, 3, 5, 1, 1, 6, 8, 1, 7, 1, 2, 3, 7, 1, 2, 1, 1, 12, 1, 1, 1, 3, 1, 1, 8, 1, 1, 2, 1, 6, 1, 1, 5, 2, 2, 3, 1, 2, 4, 4, 16, 1, 161, 45, 1, 22, 1, 2, 2, 1, 4, 1, 2, 24, 1, 2, 1, 3, 1, 2, 1, ...

Index of first occurrence of n as a [coefficient] in the continued fraction for Pi:

4, 9, 1, 30, 40, 32, 2, 44, 130, 100, 276, 55, 28, 13, 3, 78, 647, 137, 140, 180, 214, 83, 203, 91, 791, 112, 574, 175, 243, 147, 878, 455, 531, 421, 1008, 594, 784, 3041, 721, 1872, 754, 119, 492, 429, 81, 3200, 825, 283, 3027, 465, 1437, 3384, 1547, 1864, 446, ...

Of course, every integer [in the continued fraction for Pi] is not just expected to show up once, but an unlimited number of times.