In astronomy, an analemma (/ˌænəˈlɛmə/; from Greek ἀνάλημμα "support") is a diagram showing the deviation of the Sun from its mean motion in the sky, as viewed from a fixed location on the Earth. Due to the Earth's axial tilt and orbital eccentricity, the Sun will not be in the same position in the sky at the same time every day. The north–south component of the analemma is the Sun's declination, and the east–west component is the equation of time. This diagram has the form of a slender figure-eight, and can often be found on globes of the Earth.
Diagrams of analemmas frequently carry marks that show the position of the Sun at various closely spaced dates throughout the year. Analemmas with date marks can be used for various practical purposes. Without date marks, they are of little use, except as decoration.
Analemmas (as they are known today) have been used in conjunction with sundials since the 18th century to convert between apparent and mean solar time. Prior to this, the term referred to any tool or method used in the construction of sundials.
Ptolemy was the author of several scientific treatises, three of which were of continuing importance to later Byzantine, Islamic and European science. The first is the astronomical treatise now known as the Almagest, although it was originally entitled the "Mathematical Treatise" (Μαθηματικὴ Σύνταξις, Mathēmatikē Syntaxis) and then known as the "Great Treatise" (Ἡ Μεγάλη Σύνταξις, Ē Megálē Syntaxis). The second is the Geography, which is a thorough discussion of the geographic knowledge of the Greco-Roman world. This manuscript was used by Christopher Columbus as the map for his westward-bound path to Asia, in which he discovered the hitherto unknown lands of the Americas. The third is the astrological treatise in which he attempted to adapt horoscopic astrology to the Aristoteliannatural philosophy of his day. This is sometimes known as the Apotelesmatika (Ἀποτελεσματικά) but more commonly known as the Tetrabiblos from the Greek (Τετράβιβλος) meaning "Four Books" or by the Latin Quadripartitum.
The Ptolemaic Dynasty of Ancient Egypt: From Ptolemy I to Cleopatra VII
The Ptolemaic dynasty in Ancient Egypt began under the rule of Ptolemy I Soter, whose goal was to make Egypt great again and he did that by putting the country back on solid economic and administrative footing, and it ended with Cleopatra VII. When Alexander the Great of Macedon died unexpectedly in 323 BCE he had amassed an enormous empire spanning from Egypt through Anatolia, Mesopotamia and parts of India. In order to stay away from the influence of the priests and officials at Memphis, Ptolemy I moved the capital from its traditional place at Memphis to Alexandria. Alexandria, a city established by Alexander the Great, grew and flourished under the Ptolemies. The city became more of a Greek city than an Egyptian one, with the port giving easy access to the sea and Greece, and with Gree...
published: 22 Feb 2022
Ptolemy's Geocentric Model
Without modern instruments, it was very hard to understand what the planets were, especially since they seemed to wander independently from the stars. Ptolemy provided the definitive answer for the time being. Earth was the centre of the universe and all the heavenly bodies rotated around it.
For inquiry, consider the following:
1. Investigate the work of Aristarchus. What was his model of the universe and why? How did it differ from Ptolemy's?
2. Why is there no gale force wind blowing constantly as the Earth moves about the Sun? What is the cause of wind? How do hurricanes evolve?
3. What was it about the Ptolemaic model that allowed it to last for 1500 years, and what was it that eventually brought about it's demise?
4. Explain why you don't change position in a moving train when you j...
published: 03 Jul 2022
The Purpose of a System Is What It Does
The act of gaming is a gnostic ritual in which God experiences himself in small scale cosmic satire via the player's own engaged consciousness
published: 19 Sep 2024
Ptolemy I: The Founders of Egypt's Final Dynasty
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Fact Fi...
published: 31 May 2024
Overly Sarcastic Podcast: The Ptolemies
Good lord. This is a giant mess of a family tree. Just... Just watch it and see for yourself.
Downloadable audio file can be found here: http://www.mediafire.com/file/7d2fkopglxaasjb/OSPodcast+12.m4a
And all other episodes can be found here: http://www.mediafire.com/?1t64h3ehtg3pe
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published: 23 Feb 2017
A Miraculous Proof (Ptolemy's Theorem) - Numberphile
Featuring Zvezdelina Stankova... Want more?
Part 2 (bringing in Pentagons and the Golden Ratio) is at: https://youtu.be/o3QBgkQi_HA
More links & stuff in full description below ↓↓↓
Zvezda's Numberphile playlist: http://bit.ly/zvezda_videos
Zvezda's webpage: https://math.berkeley.edu/~stankova/
Epic Circles: https://youtu.be/sG_6nlMZ8f4
Triangle Magic Highway: https://youtu.be/wVH4MS6v23U
Numberphile is supported by the Mathematical Sciences Research Institute (MSRI): http://bit.ly/MSRINumberphile
We are also supported by Science Sandbox, a Simons Foundation initiative dedicated to engaging everyone with the process of science. https://www.simonsfoundation.org/outreach/science-sandbox/
And support from Math For America - https://www.mathforamerica.org/
NUMBERPHILE
Website: http://...
published: 09 Feb 2020
Ptolemy’s Theorem and the Almagest: we just found the best visual proof in 2000 years
We are making history again by presenting a new visual proof of the 2000+ years old Ptolemy's theorem and Ptolemy's inequality.
00:00 Introduction
04:27 Geometry 101
08:19 Applications
14:46 Ptolemy's inequality
18:34 LIES
25:35 Animated proofs
28:57 Thank you!
30:53 Degenerate Easter Egg
There are some other proofs of Ptolemy's theorem/inequality based on scaling and aligning suitable triangles. However, none of them is as slick, beautiful and powerful as Rainer's new proof. In particular, check out the animated scaling proof on the wiki page for Ptolemy's theorem (and this https://youtu.be/ZK08Z5A9xH4) and check out the scaling proof by Claudi Asina and Roger Nelson: Proof Without Words: Ptolemy’s Inequality in Mathematics Magazine 87, (2014), p. 291. https://www.jstor.org/stable/10.4...
published: 07 Sep 2024
033: Ptolemaic Egypt - Ptolemy I: Son of Lagus, Son of Ra
Ptolemy I Soter, a former commander of Alexander the Great turned King and Pharaoh of Egypt, was perhaps the craftiest out of all of Alexander’s Successors. A talented commander and even more brilliant politician, Ptolemy would wield diplomacy and intrigue with a deft hand, founding what would become the longest lasting Hellenistic Kingdom in his nearly 40-year reign.
Website/Episode Notes:
(https://hellenisticagepodcast.wordpress.com/2019/10/27/033-ptolemaic-egypt-ptolemy-i-son-of-lagus-son-of-ra/)
Title Theme: Seikilos Epitapth with the Lyre of Apollo, played by Lina Palera (https://soundcloud.com/user-994392473)
Support the Show:
Ko-Fi: (https://ko-fi.com/hellenisticagepodcast)
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published: 06 Jul 2020
Ludonarrative Dissidence
Chinese nationalist monkey game Black Myth: Wukong embarrasses entire Western videogame industry
The Ptolemaic dynasty in Ancient Egypt began under the rule of Ptolemy I Soter, whose goal was to make Egypt great again and he did that by putting the country ...
The Ptolemaic dynasty in Ancient Egypt began under the rule of Ptolemy I Soter, whose goal was to make Egypt great again and he did that by putting the country back on solid economic and administrative footing, and it ended with Cleopatra VII. When Alexander the Great of Macedon died unexpectedly in 323 BCE he had amassed an enormous empire spanning from Egypt through Anatolia, Mesopotamia and parts of India. In order to stay away from the influence of the priests and officials at Memphis, Ptolemy I moved the capital from its traditional place at Memphis to Alexandria. Alexandria, a city established by Alexander the Great, grew and flourished under the Ptolemies. The city became more of a Greek city than an Egyptian one, with the port giving easy access to the sea and Greece, and with Greece becoming the language of both commerce and government.
A thing to note about the Ptolemies, although they were Pharaohs of Egypt, they didn’t learn the Egyptian language and pretty much just stayed in Alexandria (which they had made the new capital, moving it from the traditional capital of Memphis). They used Greek for all of their administrative documents, and married within their own family to keep the bloodline pure and also keep their wealth in the family.
— SUPPORT US VIA OUR PATREON—
https://www.patreon.com/join/whencyclopedia
— BUY OUR MERCH —
https://www.worldhistory.store/
— CHAPTERS —
0:00 Introduction
0:51 Lead Up To The Ptolemaic Dynasty
3:50 Egypt During the Ptolemaic Dynasty
7:48 The Ptolemy Family Tree
16:49 Outro
— WANT TO KNOW MORE? —
Ptolemaic Dynasty https://www.worldhistory.org/Ptolemaic_Dynasty/
Ptolemy I https://www.worldhistory.org/Ptolemy_I/
Ptolemy XIII Theos Philopator https://www.worldhistory.org/Ptolemy_XIII_Theos_Philopator/
Arsinoe II Philadelphus https://www.worldhistory.org/Arsinoe_II_Philadelphus/
Cleopatra VII https://www.worldhistory.org/Cleopatra_VII/
The Economy of Ptolemaic Egypt https://www.worldhistory.org/article/1256/the-economy-of-ptolemaic-egypt/
Berenika II Euergetis https://www.worldhistory.org/Berenice_II_Euergetis/
— WATCH NEXT —
Cleopatra VII Philopator: the Last Queen of Ancient Egypt https://www.youtube.com/watch?v=J6bM6ElVO5k
7 Surprising Stories from Herodotus' Histories https://www.youtube.com/watch?v=QRf1jUgXRqQ
Alexander the Great: Life and Reign of the King of the Macedonian Empire https://www.youtube.com/watch?v=_E7JX-rVdFk
The Seven Wonders of the Ancient World: an Overview https://www.youtube.com/watch?v=1YWPMZ6pHm8
Comparing Two Ancient Civilisations: Ancient Egypt vs. Mesopotamia https://www.youtube.com/watch?v=EwY-ziBL1Jw
— ATTRIBUTIONS —
You can find all attribution and credits for images, animations, graphics and music here - https://worldhistory.typehut.com/the-ptolemaic-dynasty-of-ancient-egypt-from-ptolemy-i-to-cleopatra-vii-images-and-attributions-8317
The music used in this recording is the intellectual copyright of Michael Levy, a prolific composer for the recreated lyres of antiquity, and used with the creator's permission. Michael Levy's music is available to stream at all the major digital music platforms. Find out more on:
https://www.ancientlyre.com
https://open.spotify.com/artist/7Dx2vFEg8DmOJ5YCRm4A5v?si=emacIH9CRieFNGXRUyJ9
https://www.youtube.com/channel/UCJ1X6F7lGMEadnNETSzTv8A
— THUMBNAIL IMAGE —
https://www.worldhistory.org/image/15205/family-tree-of-the-ptolemaic-dynasty-of-egypt-305/
Simeon Netchev
CC BY NC SA 4.0 - https://creativecommons.org/licenses/by-nc-sa/4.0/deed.en
https://en.wikipedia.org/wiki/File:Ptolemy_I_as_Pharaoh_of_Egypt.jpg
Stella
CC BY SA 4.0 - https://creativecommons.org/licenses/by-sa/4.0/
World History Encyclopedia
www.worldhistory.org
#ptolemy #ptolemaicdynasty #cleopatravii
The Ptolemaic dynasty in Ancient Egypt began under the rule of Ptolemy I Soter, whose goal was to make Egypt great again and he did that by putting the country back on solid economic and administrative footing, and it ended with Cleopatra VII. When Alexander the Great of Macedon died unexpectedly in 323 BCE he had amassed an enormous empire spanning from Egypt through Anatolia, Mesopotamia and parts of India. In order to stay away from the influence of the priests and officials at Memphis, Ptolemy I moved the capital from its traditional place at Memphis to Alexandria. Alexandria, a city established by Alexander the Great, grew and flourished under the Ptolemies. The city became more of a Greek city than an Egyptian one, with the port giving easy access to the sea and Greece, and with Greece becoming the language of both commerce and government.
A thing to note about the Ptolemies, although they were Pharaohs of Egypt, they didn’t learn the Egyptian language and pretty much just stayed in Alexandria (which they had made the new capital, moving it from the traditional capital of Memphis). They used Greek for all of their administrative documents, and married within their own family to keep the bloodline pure and also keep their wealth in the family.
— SUPPORT US VIA OUR PATREON—
https://www.patreon.com/join/whencyclopedia
— BUY OUR MERCH —
https://www.worldhistory.store/
— CHAPTERS —
0:00 Introduction
0:51 Lead Up To The Ptolemaic Dynasty
3:50 Egypt During the Ptolemaic Dynasty
7:48 The Ptolemy Family Tree
16:49 Outro
— WANT TO KNOW MORE? —
Ptolemaic Dynasty https://www.worldhistory.org/Ptolemaic_Dynasty/
Ptolemy I https://www.worldhistory.org/Ptolemy_I/
Ptolemy XIII Theos Philopator https://www.worldhistory.org/Ptolemy_XIII_Theos_Philopator/
Arsinoe II Philadelphus https://www.worldhistory.org/Arsinoe_II_Philadelphus/
Cleopatra VII https://www.worldhistory.org/Cleopatra_VII/
The Economy of Ptolemaic Egypt https://www.worldhistory.org/article/1256/the-economy-of-ptolemaic-egypt/
Berenika II Euergetis https://www.worldhistory.org/Berenice_II_Euergetis/
— WATCH NEXT —
Cleopatra VII Philopator: the Last Queen of Ancient Egypt https://www.youtube.com/watch?v=J6bM6ElVO5k
7 Surprising Stories from Herodotus' Histories https://www.youtube.com/watch?v=QRf1jUgXRqQ
Alexander the Great: Life and Reign of the King of the Macedonian Empire https://www.youtube.com/watch?v=_E7JX-rVdFk
The Seven Wonders of the Ancient World: an Overview https://www.youtube.com/watch?v=1YWPMZ6pHm8
Comparing Two Ancient Civilisations: Ancient Egypt vs. Mesopotamia https://www.youtube.com/watch?v=EwY-ziBL1Jw
— ATTRIBUTIONS —
You can find all attribution and credits for images, animations, graphics and music here - https://worldhistory.typehut.com/the-ptolemaic-dynasty-of-ancient-egypt-from-ptolemy-i-to-cleopatra-vii-images-and-attributions-8317
The music used in this recording is the intellectual copyright of Michael Levy, a prolific composer for the recreated lyres of antiquity, and used with the creator's permission. Michael Levy's music is available to stream at all the major digital music platforms. Find out more on:
https://www.ancientlyre.com
https://open.spotify.com/artist/7Dx2vFEg8DmOJ5YCRm4A5v?si=emacIH9CRieFNGXRUyJ9
https://www.youtube.com/channel/UCJ1X6F7lGMEadnNETSzTv8A
— THUMBNAIL IMAGE —
https://www.worldhistory.org/image/15205/family-tree-of-the-ptolemaic-dynasty-of-egypt-305/
Simeon Netchev
CC BY NC SA 4.0 - https://creativecommons.org/licenses/by-nc-sa/4.0/deed.en
https://en.wikipedia.org/wiki/File:Ptolemy_I_as_Pharaoh_of_Egypt.jpg
Stella
CC BY SA 4.0 - https://creativecommons.org/licenses/by-sa/4.0/
World History Encyclopedia
www.worldhistory.org
#ptolemy #ptolemaicdynasty #cleopatravii
Without modern instruments, it was very hard to understand what the planets were, especially since they seemed to wander independently from the stars. Ptolemy p...
Without modern instruments, it was very hard to understand what the planets were, especially since they seemed to wander independently from the stars. Ptolemy provided the definitive answer for the time being. Earth was the centre of the universe and all the heavenly bodies rotated around it.
For inquiry, consider the following:
1. Investigate the work of Aristarchus. What was his model of the universe and why? How did it differ from Ptolemy's?
2. Why is there no gale force wind blowing constantly as the Earth moves about the Sun? What is the cause of wind? How do hurricanes evolve?
3. What was it about the Ptolemaic model that allowed it to last for 1500 years, and what was it that eventually brought about it's demise?
4. Explain why you don't change position in a moving train when you jump up in the air. Why does the train not move out from under you?
5. Use this website as a starting point to do some research and reporting into the great minds that have influenced our understanding of the heavens: https://starchild.gsfc.nasa.gov/docs/StarChild/universe_level2/cosmology.html
6. In Western society, we often hear only about the Greeks and perhaps the Egyptians, with regard to astronomy. Find one major astronomical contribution made by the Chinese, the Indian and at least one First Nation.
Without modern instruments, it was very hard to understand what the planets were, especially since they seemed to wander independently from the stars. Ptolemy provided the definitive answer for the time being. Earth was the centre of the universe and all the heavenly bodies rotated around it.
For inquiry, consider the following:
1. Investigate the work of Aristarchus. What was his model of the universe and why? How did it differ from Ptolemy's?
2. Why is there no gale force wind blowing constantly as the Earth moves about the Sun? What is the cause of wind? How do hurricanes evolve?
3. What was it about the Ptolemaic model that allowed it to last for 1500 years, and what was it that eventually brought about it's demise?
4. Explain why you don't change position in a moving train when you jump up in the air. Why does the train not move out from under you?
5. Use this website as a starting point to do some research and reporting into the great minds that have influenced our understanding of the heavens: https://starchild.gsfc.nasa.gov/docs/StarChild/universe_level2/cosmology.html
6. In Western society, we often hear only about the Greeks and perhaps the Egyptians, with regard to astronomy. Find one major astronomical contribution made by the Chinese, the Indian and at least one First Nation.
Head to Squarespace.com for a free trial, and when you're ready to launch, go to http://squarespace.com/biographics to save 10% off on your first purchase of a ...
Head to Squarespace.com for a free trial, and when you're ready to launch, go to http://squarespace.com/biographics to save 10% off on your first purchase of a website/domain using the code BIOGRAPHICS
→ Subscribe for new videos at least twice a week!
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Love content? Check out our other YouTube Channels:
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Storyrant: https://www.youtube.com/@UC6Sk2h2WsYMxnGERAH_FbGg
OEI analog horror experiment: https://www.youtube.com/@UCsz5l0hY67TGoJR0CZiWVnA
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Further Reading:
Samuel Sharpe - The History of Egypt under the Ptolemies
C. A. Kincaid - Successors of Alexander the Great
Ian Shaw - The Oxford History of Ancient Egypt
Diodorus Siculus - Library of History
Livius
WorldHistory
Head to Squarespace.com for a free trial, and when you're ready to launch, go to http://squarespace.com/biographics to save 10% off on your first purchase of a website/domain using the code BIOGRAPHICS
→ Subscribe for new videos at least twice a week!
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Love content? Check out our other YouTube Channels:
Geographics: https://www.youtube.com/channel/UCHKRfxkMTqiiv4pF99qGKIw
TopTenz: https://www.youtube.com/user/toptenznet
Eric's Stuff:
Storyrant: https://www.youtube.com/@UC6Sk2h2WsYMxnGERAH_FbGg
OEI analog horror experiment: https://www.youtube.com/@UCsz5l0hY67TGoJR0CZiWVnA
Free Scifi-Horror Ebook: https://dl.bookfunnel.com/1cw07o2uyb
Link Tree to Other Books and Socials: https://linktr.ee/EricMalikyte
Karl's Socials:
Fact Fiend: https://www.youtube.com/@FactFiend
Wiki Weekends: https://www.youtube.com/@wikiweekends
Untitled Side Channel: https://www.youtube.com/@untitledsidechannel
Twitter: https://twitter.com/KarlSmallwood
Further Reading:
Samuel Sharpe - The History of Egypt under the Ptolemies
C. A. Kincaid - Successors of Alexander the Great
Ian Shaw - The Oxford History of Ancient Egypt
Diodorus Siculus - Library of History
Livius
WorldHistory
Good lord. This is a giant mess of a family tree. Just... Just watch it and see for yourself.
Downloadable audio file can be found here: http://www.mediafire.c...
Good lord. This is a giant mess of a family tree. Just... Just watch it and see for yourself.
Downloadable audio file can be found here: http://www.mediafire.com/file/7d2fkopglxaasjb/OSPodcast+12.m4a
And all other episodes can be found here: http://www.mediafire.com/?1t64h3ehtg3pe
PATREON: www.patreon.com/user?u=4664797
MERCH LINKS:
Shirts - https://overlysarcasticproducts.threadless.com/designs
All the other stuff - http://www.cafepress.com/OverlySarcasticProducts
Find us on Twitter @OSPYouTube!
Good lord. This is a giant mess of a family tree. Just... Just watch it and see for yourself.
Downloadable audio file can be found here: http://www.mediafire.com/file/7d2fkopglxaasjb/OSPodcast+12.m4a
And all other episodes can be found here: http://www.mediafire.com/?1t64h3ehtg3pe
PATREON: www.patreon.com/user?u=4664797
MERCH LINKS:
Shirts - https://overlysarcasticproducts.threadless.com/designs
All the other stuff - http://www.cafepress.com/OverlySarcasticProducts
Find us on Twitter @OSPYouTube!
Featuring Zvezdelina Stankova... Want more?
Part 2 (bringing in Pentagons and the Golden Ratio) is at: https://youtu.be/o3QBgkQi_HA
More links & stuff in full ...
Featuring Zvezdelina Stankova... Want more?
Part 2 (bringing in Pentagons and the Golden Ratio) is at: https://youtu.be/o3QBgkQi_HA
More links & stuff in full description below ↓↓↓
Zvezda's Numberphile playlist: http://bit.ly/zvezda_videos
Zvezda's webpage: https://math.berkeley.edu/~stankova/
Epic Circles: https://youtu.be/sG_6nlMZ8f4
Triangle Magic Highway: https://youtu.be/wVH4MS6v23U
Numberphile is supported by the Mathematical Sciences Research Institute (MSRI): http://bit.ly/MSRINumberphile
We are also supported by Science Sandbox, a Simons Foundation initiative dedicated to engaging everyone with the process of science. https://www.simonsfoundation.org/outreach/science-sandbox/
And support from Math For America - https://www.mathforamerica.org/
NUMBERPHILE
Website: http://www.numberphile.com/
Numberphile on Facebook: http://www.facebook.com/numberphile
Numberphile tweets: https://twitter.com/numberphile
Subscribe: http://bit.ly/Numberphile_Sub
Videos by Brady Haran
Additional editing and animation by Pete McPartlan
Patreon: http://www.patreon.com/numberphile
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Brady's videos subreddit: http://www.reddit.com/r/BradyHaran/
Brady's latest videos across all channels: http://www.bradyharanblog.com/
Sign up for (occasional) emails: http://eepurl.com/YdjL9
Featuring Zvezdelina Stankova... Want more?
Part 2 (bringing in Pentagons and the Golden Ratio) is at: https://youtu.be/o3QBgkQi_HA
More links & stuff in full description below ↓↓↓
Zvezda's Numberphile playlist: http://bit.ly/zvezda_videos
Zvezda's webpage: https://math.berkeley.edu/~stankova/
Epic Circles: https://youtu.be/sG_6nlMZ8f4
Triangle Magic Highway: https://youtu.be/wVH4MS6v23U
Numberphile is supported by the Mathematical Sciences Research Institute (MSRI): http://bit.ly/MSRINumberphile
We are also supported by Science Sandbox, a Simons Foundation initiative dedicated to engaging everyone with the process of science. https://www.simonsfoundation.org/outreach/science-sandbox/
And support from Math For America - https://www.mathforamerica.org/
NUMBERPHILE
Website: http://www.numberphile.com/
Numberphile on Facebook: http://www.facebook.com/numberphile
Numberphile tweets: https://twitter.com/numberphile
Subscribe: http://bit.ly/Numberphile_Sub
Videos by Brady Haran
Additional editing and animation by Pete McPartlan
Patreon: http://www.patreon.com/numberphile
Numberphile T-Shirts and Merch: https://teespring.com/stores/numberphile
Brady's videos subreddit: http://www.reddit.com/r/BradyHaran/
Brady's latest videos across all channels: http://www.bradyharanblog.com/
Sign up for (occasional) emails: http://eepurl.com/YdjL9
We are making history again by presenting a new visual proof of the 2000+ years old Ptolemy's theorem and Ptolemy's inequality.
00:00 Introduction
04:27 Geomet...
We are making history again by presenting a new visual proof of the 2000+ years old Ptolemy's theorem and Ptolemy's inequality.
00:00 Introduction
04:27 Geometry 101
08:19 Applications
14:46 Ptolemy's inequality
18:34 LIES
25:35 Animated proofs
28:57 Thank you!
30:53 Degenerate Easter Egg
There are some other proofs of Ptolemy's theorem/inequality based on scaling and aligning suitable triangles. However, none of them is as slick, beautiful and powerful as Rainer's new proof. In particular, check out the animated scaling proof on the wiki page for Ptolemy's theorem (and this https://youtu.be/ZK08Z5A9xH4) and check out the scaling proof by Claudi Asina and Roger Nelson: Proof Without Words: Ptolemy’s Inequality in Mathematics Magazine 87, (2014), p. 291. https://www.jstor.org/stable/10.4169/math.mag.87.4.291
Rainer was inspired by a classic scaling based proof of Pythagoras theorem that I presented here https://youtu.be/p-0SOWbzUYI?si=GeGzZ0R_Dj1AsXqR&t=371
You can find a couple of full text versions of the Almagest here
https://www.wilbourhall.org/index.html#ptolemy
https://classicalliberalarts.com/resources/PTOLEMY_ALMAGEST_ENGLISH.pdf
For more background info check out the very comprehensive wiki pages on:
Ptolemy’s theorem
https://en.wikipedia.org/wiki/Ptolemy%27s_theorem
Ptolemy’s inequality
https://en.wikipedia.org/wiki/Ptolemy%27s_inequality
Claudius Ptolemy
https://en.wikipedia.org/wiki/Ptolemy
The Almagest
https://sco.wikipedia.org/wiki/Almagest
Trigonometric identities
https://en.wikipedia.org/wiki/List_of_trigonometric_identities
Cyclic quadrilateral
https://en.wikipedia.org/wiki/Cyclic_quadrilateral
The optic equation
https://en.wikipedia.org/wiki/Optic_equation
There are very interesting higher-dimensional versions of Ptolemy's theorem just like there are higher-dimensional versions of Pythagoras theorem. I did not get around to talking them today. Google ...
Highly recommended:
T. Brendan, How Ptolemy constructed trigonometry tables, The Mathematics Teacher 58 (1965), pp. 141-149 https://www.jstor.org/stable/27967990
Tom M. Apostol, Ptolemy's Inequality and the Chordal Metric, Mathematics Magazine 40 (1967), pp. 233-235 https://www.jstor.org/stable/2688275
https://demonstrations.wolfram.com/PtolemysTableOfChords/ an interactive exploration of Ptolemy's table of chords
Ptolemy's theorem made a guest appearance in the the previous Mathologer video on the golden ratio: https://youtu.be/cCXRUHUgvLI
Here is a nice trick to make Ptolemy counterparts of Pythagorean triples. Take any two sets of Pythagorean triples:
5² = 3² + 4², 13² = 12² + 5², and combine them like this:
65² = 13² × 5²= 13²(4² + 3²) = 52² + 39²= 5²(12² + 5²) = 60² + 25².
Now combining the two right angled triangles 52-39-65 and 25-60-65 along the common diagonal in any of four different ways gives a convex quadrilateral with all sides integers. Note that you automatically get 5 integer lengths and then Ptolemy's theorem guarantees that the remaining side is a fraction. Scaling up everything by the denominator of that fraction gives one of the special integer-everywhere quadrilaterals. See also Brahmagupta quadrilaterals.
Here is a nice application of Ptolemy's theorem to a International Maths Olympiad problem https://www.youtube.com/watch?v=NHjtHOE1lks
In a cyclic quadrilateral the ratio of the diagonals equals the ratio of the sums of products of the sides that share the diagonals' end points: https://www.geogebra.org/m/XQr5jJQg This extension of Ptolemy's theorem is part of the thumbnail for this video.
T-shirt: cowsine :)
Music: Floating branch by Muted and I promise by Ian Post.
Enjoy,
burkard
We are making history again by presenting a new visual proof of the 2000+ years old Ptolemy's theorem and Ptolemy's inequality.
00:00 Introduction
04:27 Geometry 101
08:19 Applications
14:46 Ptolemy's inequality
18:34 LIES
25:35 Animated proofs
28:57 Thank you!
30:53 Degenerate Easter Egg
There are some other proofs of Ptolemy's theorem/inequality based on scaling and aligning suitable triangles. However, none of them is as slick, beautiful and powerful as Rainer's new proof. In particular, check out the animated scaling proof on the wiki page for Ptolemy's theorem (and this https://youtu.be/ZK08Z5A9xH4) and check out the scaling proof by Claudi Asina and Roger Nelson: Proof Without Words: Ptolemy’s Inequality in Mathematics Magazine 87, (2014), p. 291. https://www.jstor.org/stable/10.4169/math.mag.87.4.291
Rainer was inspired by a classic scaling based proof of Pythagoras theorem that I presented here https://youtu.be/p-0SOWbzUYI?si=GeGzZ0R_Dj1AsXqR&t=371
You can find a couple of full text versions of the Almagest here
https://www.wilbourhall.org/index.html#ptolemy
https://classicalliberalarts.com/resources/PTOLEMY_ALMAGEST_ENGLISH.pdf
For more background info check out the very comprehensive wiki pages on:
Ptolemy’s theorem
https://en.wikipedia.org/wiki/Ptolemy%27s_theorem
Ptolemy’s inequality
https://en.wikipedia.org/wiki/Ptolemy%27s_inequality
Claudius Ptolemy
https://en.wikipedia.org/wiki/Ptolemy
The Almagest
https://sco.wikipedia.org/wiki/Almagest
Trigonometric identities
https://en.wikipedia.org/wiki/List_of_trigonometric_identities
Cyclic quadrilateral
https://en.wikipedia.org/wiki/Cyclic_quadrilateral
The optic equation
https://en.wikipedia.org/wiki/Optic_equation
There are very interesting higher-dimensional versions of Ptolemy's theorem just like there are higher-dimensional versions of Pythagoras theorem. I did not get around to talking them today. Google ...
Highly recommended:
T. Brendan, How Ptolemy constructed trigonometry tables, The Mathematics Teacher 58 (1965), pp. 141-149 https://www.jstor.org/stable/27967990
Tom M. Apostol, Ptolemy's Inequality and the Chordal Metric, Mathematics Magazine 40 (1967), pp. 233-235 https://www.jstor.org/stable/2688275
https://demonstrations.wolfram.com/PtolemysTableOfChords/ an interactive exploration of Ptolemy's table of chords
Ptolemy's theorem made a guest appearance in the the previous Mathologer video on the golden ratio: https://youtu.be/cCXRUHUgvLI
Here is a nice trick to make Ptolemy counterparts of Pythagorean triples. Take any two sets of Pythagorean triples:
5² = 3² + 4², 13² = 12² + 5², and combine them like this:
65² = 13² × 5²= 13²(4² + 3²) = 52² + 39²= 5²(12² + 5²) = 60² + 25².
Now combining the two right angled triangles 52-39-65 and 25-60-65 along the common diagonal in any of four different ways gives a convex quadrilateral with all sides integers. Note that you automatically get 5 integer lengths and then Ptolemy's theorem guarantees that the remaining side is a fraction. Scaling up everything by the denominator of that fraction gives one of the special integer-everywhere quadrilaterals. See also Brahmagupta quadrilaterals.
Here is a nice application of Ptolemy's theorem to a International Maths Olympiad problem https://www.youtube.com/watch?v=NHjtHOE1lks
In a cyclic quadrilateral the ratio of the diagonals equals the ratio of the sums of products of the sides that share the diagonals' end points: https://www.geogebra.org/m/XQr5jJQg This extension of Ptolemy's theorem is part of the thumbnail for this video.
T-shirt: cowsine :)
Music: Floating branch by Muted and I promise by Ian Post.
Enjoy,
burkard
Ptolemy I Soter, a former commander of Alexander the Great turned King and Pharaoh of Egypt, was perhaps the craftiest out of all of Alexander’s Successors. A t...
Ptolemy I Soter, a former commander of Alexander the Great turned King and Pharaoh of Egypt, was perhaps the craftiest out of all of Alexander’s Successors. A talented commander and even more brilliant politician, Ptolemy would wield diplomacy and intrigue with a deft hand, founding what would become the longest lasting Hellenistic Kingdom in his nearly 40-year reign.
Website/Episode Notes:
(https://hellenisticagepodcast.wordpress.com/2019/10/27/033-ptolemaic-egypt-ptolemy-i-son-of-lagus-son-of-ra/)
Title Theme: Seikilos Epitapth with the Lyre of Apollo, played by Lina Palera (https://soundcloud.com/user-994392473)
Support the Show:
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Social Media:
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Ptolemy I Soter, a former commander of Alexander the Great turned King and Pharaoh of Egypt, was perhaps the craftiest out of all of Alexander’s Successors. A talented commander and even more brilliant politician, Ptolemy would wield diplomacy and intrigue with a deft hand, founding what would become the longest lasting Hellenistic Kingdom in his nearly 40-year reign.
Website/Episode Notes:
(https://hellenisticagepodcast.wordpress.com/2019/10/27/033-ptolemaic-egypt-ptolemy-i-son-of-lagus-son-of-ra/)
Title Theme: Seikilos Epitapth with the Lyre of Apollo, played by Lina Palera (https://soundcloud.com/user-994392473)
Support the Show:
Ko-Fi: (https://ko-fi.com/hellenisticagepodcast)
Etsy: (https://www.etsy.com/shop/Hellenistic...)
Amazon Wishlist: (https://www.amazon.com/hz/wishlist/ls...)
Social Media:
Twitter (https://twitter.com/HellenisticPod)
Facebook (www.facebook.com/hellenisticagepodcast/)
Discord (https://discord.gg/VJcyUcN)
The Ptolemaic dynasty in Ancient Egypt began under the rule of Ptolemy I Soter, whose goal was to make Egypt great again and he did that by putting the country back on solid economic and administrative footing, and it ended with Cleopatra VII. When Alexander the Great of Macedon died unexpectedly in 323 BCE he had amassed an enormous empire spanning from Egypt through Anatolia, Mesopotamia and parts of India. In order to stay away from the influence of the priests and officials at Memphis, Ptolemy I moved the capital from its traditional place at Memphis to Alexandria. Alexandria, a city established by Alexander the Great, grew and flourished under the Ptolemies. The city became more of a Greek city than an Egyptian one, with the port giving easy access to the sea and Greece, and with Greece becoming the language of both commerce and government.
A thing to note about the Ptolemies, although they were Pharaohs of Egypt, they didn’t learn the Egyptian language and pretty much just stayed in Alexandria (which they had made the new capital, moving it from the traditional capital of Memphis). They used Greek for all of their administrative documents, and married within their own family to keep the bloodline pure and also keep their wealth in the family.
— SUPPORT US VIA OUR PATREON—
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— CHAPTERS —
0:00 Introduction
0:51 Lead Up To The Ptolemaic Dynasty
3:50 Egypt During the Ptolemaic Dynasty
7:48 The Ptolemy Family Tree
16:49 Outro
— WANT TO KNOW MORE? —
Ptolemaic Dynasty https://www.worldhistory.org/Ptolemaic_Dynasty/
Ptolemy I https://www.worldhistory.org/Ptolemy_I/
Ptolemy XIII Theos Philopator https://www.worldhistory.org/Ptolemy_XIII_Theos_Philopator/
Arsinoe II Philadelphus https://www.worldhistory.org/Arsinoe_II_Philadelphus/
Cleopatra VII https://www.worldhistory.org/Cleopatra_VII/
The Economy of Ptolemaic Egypt https://www.worldhistory.org/article/1256/the-economy-of-ptolemaic-egypt/
Berenika II Euergetis https://www.worldhistory.org/Berenice_II_Euergetis/
— WATCH NEXT —
Cleopatra VII Philopator: the Last Queen of Ancient Egypt https://www.youtube.com/watch?v=J6bM6ElVO5k
7 Surprising Stories from Herodotus' Histories https://www.youtube.com/watch?v=QRf1jUgXRqQ
Alexander the Great: Life and Reign of the King of the Macedonian Empire https://www.youtube.com/watch?v=_E7JX-rVdFk
The Seven Wonders of the Ancient World: an Overview https://www.youtube.com/watch?v=1YWPMZ6pHm8
Comparing Two Ancient Civilisations: Ancient Egypt vs. Mesopotamia https://www.youtube.com/watch?v=EwY-ziBL1Jw
— ATTRIBUTIONS —
You can find all attribution and credits for images, animations, graphics and music here - https://worldhistory.typehut.com/the-ptolemaic-dynasty-of-ancient-egypt-from-ptolemy-i-to-cleopatra-vii-images-and-attributions-8317
The music used in this recording is the intellectual copyright of Michael Levy, a prolific composer for the recreated lyres of antiquity, and used with the creator's permission. Michael Levy's music is available to stream at all the major digital music platforms. Find out more on:
https://www.ancientlyre.com
https://open.spotify.com/artist/7Dx2vFEg8DmOJ5YCRm4A5v?si=emacIH9CRieFNGXRUyJ9
https://www.youtube.com/channel/UCJ1X6F7lGMEadnNETSzTv8A
— THUMBNAIL IMAGE —
https://www.worldhistory.org/image/15205/family-tree-of-the-ptolemaic-dynasty-of-egypt-305/
Simeon Netchev
CC BY NC SA 4.0 - https://creativecommons.org/licenses/by-nc-sa/4.0/deed.en
https://en.wikipedia.org/wiki/File:Ptolemy_I_as_Pharaoh_of_Egypt.jpg
Stella
CC BY SA 4.0 - https://creativecommons.org/licenses/by-sa/4.0/
World History Encyclopedia
www.worldhistory.org
#ptolemy #ptolemaicdynasty #cleopatravii
Without modern instruments, it was very hard to understand what the planets were, especially since they seemed to wander independently from the stars. Ptolemy provided the definitive answer for the time being. Earth was the centre of the universe and all the heavenly bodies rotated around it.
For inquiry, consider the following:
1. Investigate the work of Aristarchus. What was his model of the universe and why? How did it differ from Ptolemy's?
2. Why is there no gale force wind blowing constantly as the Earth moves about the Sun? What is the cause of wind? How do hurricanes evolve?
3. What was it about the Ptolemaic model that allowed it to last for 1500 years, and what was it that eventually brought about it's demise?
4. Explain why you don't change position in a moving train when you jump up in the air. Why does the train not move out from under you?
5. Use this website as a starting point to do some research and reporting into the great minds that have influenced our understanding of the heavens: https://starchild.gsfc.nasa.gov/docs/StarChild/universe_level2/cosmology.html
6. In Western society, we often hear only about the Greeks and perhaps the Egyptians, with regard to astronomy. Find one major astronomical contribution made by the Chinese, the Indian and at least one First Nation.
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Further Reading:
Samuel Sharpe - The History of Egypt under the Ptolemies
C. A. Kincaid - Successors of Alexander the Great
Ian Shaw - The Oxford History of Ancient Egypt
Diodorus Siculus - Library of History
Livius
WorldHistory
Good lord. This is a giant mess of a family tree. Just... Just watch it and see for yourself.
Downloadable audio file can be found here: http://www.mediafire.com/file/7d2fkopglxaasjb/OSPodcast+12.m4a
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We are making history again by presenting a new visual proof of the 2000+ years old Ptolemy's theorem and Ptolemy's inequality.
00:00 Introduction
04:27 Geometry 101
08:19 Applications
14:46 Ptolemy's inequality
18:34 LIES
25:35 Animated proofs
28:57 Thank you!
30:53 Degenerate Easter Egg
There are some other proofs of Ptolemy's theorem/inequality based on scaling and aligning suitable triangles. However, none of them is as slick, beautiful and powerful as Rainer's new proof. In particular, check out the animated scaling proof on the wiki page for Ptolemy's theorem (and this https://youtu.be/ZK08Z5A9xH4) and check out the scaling proof by Claudi Asina and Roger Nelson: Proof Without Words: Ptolemy’s Inequality in Mathematics Magazine 87, (2014), p. 291. https://www.jstor.org/stable/10.4169/math.mag.87.4.291
Rainer was inspired by a classic scaling based proof of Pythagoras theorem that I presented here https://youtu.be/p-0SOWbzUYI?si=GeGzZ0R_Dj1AsXqR&t=371
You can find a couple of full text versions of the Almagest here
https://www.wilbourhall.org/index.html#ptolemy
https://classicalliberalarts.com/resources/PTOLEMY_ALMAGEST_ENGLISH.pdf
For more background info check out the very comprehensive wiki pages on:
Ptolemy’s theorem
https://en.wikipedia.org/wiki/Ptolemy%27s_theorem
Ptolemy’s inequality
https://en.wikipedia.org/wiki/Ptolemy%27s_inequality
Claudius Ptolemy
https://en.wikipedia.org/wiki/Ptolemy
The Almagest
https://sco.wikipedia.org/wiki/Almagest
Trigonometric identities
https://en.wikipedia.org/wiki/List_of_trigonometric_identities
Cyclic quadrilateral
https://en.wikipedia.org/wiki/Cyclic_quadrilateral
The optic equation
https://en.wikipedia.org/wiki/Optic_equation
There are very interesting higher-dimensional versions of Ptolemy's theorem just like there are higher-dimensional versions of Pythagoras theorem. I did not get around to talking them today. Google ...
Highly recommended:
T. Brendan, How Ptolemy constructed trigonometry tables, The Mathematics Teacher 58 (1965), pp. 141-149 https://www.jstor.org/stable/27967990
Tom M. Apostol, Ptolemy's Inequality and the Chordal Metric, Mathematics Magazine 40 (1967), pp. 233-235 https://www.jstor.org/stable/2688275
https://demonstrations.wolfram.com/PtolemysTableOfChords/ an interactive exploration of Ptolemy's table of chords
Ptolemy's theorem made a guest appearance in the the previous Mathologer video on the golden ratio: https://youtu.be/cCXRUHUgvLI
Here is a nice trick to make Ptolemy counterparts of Pythagorean triples. Take any two sets of Pythagorean triples:
5² = 3² + 4², 13² = 12² + 5², and combine them like this:
65² = 13² × 5²= 13²(4² + 3²) = 52² + 39²= 5²(12² + 5²) = 60² + 25².
Now combining the two right angled triangles 52-39-65 and 25-60-65 along the common diagonal in any of four different ways gives a convex quadrilateral with all sides integers. Note that you automatically get 5 integer lengths and then Ptolemy's theorem guarantees that the remaining side is a fraction. Scaling up everything by the denominator of that fraction gives one of the special integer-everywhere quadrilaterals. See also Brahmagupta quadrilaterals.
Here is a nice application of Ptolemy's theorem to a International Maths Olympiad problem https://www.youtube.com/watch?v=NHjtHOE1lks
In a cyclic quadrilateral the ratio of the diagonals equals the ratio of the sums of products of the sides that share the diagonals' end points: https://www.geogebra.org/m/XQr5jJQg This extension of Ptolemy's theorem is part of the thumbnail for this video.
T-shirt: cowsine :)
Music: Floating branch by Muted and I promise by Ian Post.
Enjoy,
burkard
Ptolemy I Soter, a former commander of Alexander the Great turned King and Pharaoh of Egypt, was perhaps the craftiest out of all of Alexander’s Successors. A talented commander and even more brilliant politician, Ptolemy would wield diplomacy and intrigue with a deft hand, founding what would become the longest lasting Hellenistic Kingdom in his nearly 40-year reign.
Website/Episode Notes:
(https://hellenisticagepodcast.wordpress.com/2019/10/27/033-ptolemaic-egypt-ptolemy-i-son-of-lagus-son-of-ra/)
Title Theme: Seikilos Epitapth with the Lyre of Apollo, played by Lina Palera (https://soundcloud.com/user-994392473)
Support the Show:
Ko-Fi: (https://ko-fi.com/hellenisticagepodcast)
Etsy: (https://www.etsy.com/shop/Hellenistic...)
Amazon Wishlist: (https://www.amazon.com/hz/wishlist/ls...)
Social Media:
Twitter (https://twitter.com/HellenisticPod)
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In astronomy, an analemma (/ˌænəˈlɛmə/; from Greek ἀνάλημμα "support") is a diagram showing the deviation of the Sun from its mean motion in the sky, as viewed from a fixed location on the Earth. Due to the Earth's axial tilt and orbital eccentricity, the Sun will not be in the same position in the sky at the same time every day. The north–south component of the analemma is the Sun's declination, and the east–west component is the equation of time. This diagram has the form of a slender figure-eight, and can often be found on globes of the Earth.
Diagrams of analemmas frequently carry marks that show the position of the Sun at various closely spaced dates throughout the year. Analemmas with date marks can be used for various practical purposes. Without date marks, they are of little use, except as decoration.
Analemmas (as they are known today) have been used in conjunction with sundials since the 18th century to convert between apparent and mean solar time. Prior to this, the term referred to any tool or method used in the construction of sundials.