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-
Apollonius of Perga
Apollonius was an influential Greek mathematician and astronomer, born in a region of what is now Turkey. He became known as the "Great Geometer."
My other YouTube channels:
The Science Fiction Rock Experience ( the music show I produce):
https://www.youtube.com/@ScienceFictionRockExperience
My science and music channel::
https://www.youtube.com/@drdaviddarling
Science World (with Emrah Polat):
https://www.youtube.com/@ScienceWorld1
My website: https://www.daviddarling.info
My latest book is available here: https://oneworld-publications.com/work/the-biggest-number-in-the-world/
published: 05 Oct 2022
-
Apollonius of Perga
Dr Sergis discusses the contributions of another mathematician, Apollonius of Perga.
*************
The Dr Sergis Academy is a highly respected tutoring centre based in Enfield, North London. 🧡
For more information, please visit www.dsacademy.co.uk
We are also on Facebook, Instagram, TikTok and X, just search for the Dr Sergis Academy. 🔎
#enfield #thedrsergisacademy #mathematician #apolloniusofperga #tuition #tutors #ks2 #ks3 #gcse #alevel #english #maths #science
published: 09 Dec 2023
-
Apollonius of Perga
published: 24 Mar 2014
-
Apollonius Cone
Apollonius of Perga was a Greek mathematician who lived from approximately 260 BC to 190 BC. One of his major written works, parts of which still exist, dealt with what is termed Conic Sections. His work is said to have influenced many later scholars, including Isaac Newton and René Descartes.
Conic Sections deals with the shapes that you get when a cone is intersected by a plane. Essentially you can get one of four shapes, a circle, an ellipse, a parabola or a hyperbola. It is believed to be Apollonius who gave the last three shapes the names we use today.
There was a time when most children in school learned about conic sections, although it is not so common nowadays.
This beautiful hand made wooden version of the Apollonius Cone stands 27 cm high. It comes apart into five pieces, and...
published: 01 Mar 2012
-
Apollonius' circle construction problems | Famous Math Problems 3 | NJ Wildberger
Around 200 B.C., Apollonius of Perga, the greatest geometer of all time, gave a series of related problems; how to construct a circle in the plane touching three objects, where the objects are either a point (P), a line (L) and or a circle (C). Many mathematicians have studied this most famous of all geometric problems.
In this video we give some of the history, discuss Euclid's solution to two of the problems (which predate Apollonius' work), and summarize some circle geometry that is relevant to the subject: the Subtended spreads theorem from Rational Trigonometry, the power of a point wrt a circle, the pole-polar duality determined between points and lines by a circle, and inversion.
We also discuss the importance of constructions in both education and theoretical work---a point which...
published: 23 Jan 2013
-
Problem of Apollonius - what does it teach us about problem solving?
This video uses the problem of Apollonius as a way to introduce circle inversion and an important problem-solving technique - transforming a hard problem into a simpler one; then solve for the simpler, transformed version of the problem before doing the inverse transformation so that we obtain the solution to the hard problem. This problem-solving technique is the motive behind Fourier transform and other transforms like that (Laplace / Mellin): transforming the forced ODEs to a polynomial function, which is much easier than the original problem.
Circle inversion is also an important technique in Euclidean geometry - it grants access to many more advanced geometric problems, like Pappus chain. This is because it really changes the form of the geometric object as opposed to translation, re...
published: 21 Sep 2020
-
Apollonius and polarity | Universal Hyperbolic Geometry 1 | NJ Wildberger
This is the start of a new course on hyperbolic geometry that features a revolutionary simplifed approach to the subject, framing it in terms of classical projective geometry and the study of a distinguished circle. This subject will be called Universal Hyperbolic Geometry, as it extends the subject to arbitrary fields, as well as to the outside of the light cone/null circle.
We begin by going back to Apollonius of Perga (in present day Turkey, not Italy!) and his understanding of the crucial role of polarity in studying conics, in particular the circle. Given a fixed circle, to each point in the plane we associate a line called the polar, and conversely to a line we associated a point called its pole. This duality is all important for hyperbolic geometry
CONTENT SUMMARY: conics @00:00 t...
published: 18 Apr 2011
-
Apollonius of Perga | Heroes of Mathematics | Mathematicians
Discover the profound mathematical contributions of Apollonius of Perga in this enlightening video. Apollonius, a mathematical luminary from the 3rd century BCE, revolutionized the study of conic sections, encompassing circles, ellipses, parabolas, and hyperbolas. His groundbreaking theorems and concepts continue to shape modern mathematics and various scientific fields. Join us as we delve into Apollonius's life and his enduring legacy, offering a glimpse into the ancient world of mathematics. Whether you're a math enthusiast, history buff, or simply curious, this video provides a captivating journey through the foundations of geometry. Don't miss out – like, subscribe, and share to spread the brilliance of Apollonius of Perga and his timeless contributions to mathematics and science.
Th...
published: 13 Sep 2023
-
Precal Spring 22 Apollonius Parabola
published: 24 Jan 2020
-
Apollonius of Perga | Wikipedia audio article
This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Apollonius_of_Perga
00:00:50 1 Life
00:04:34 1.1 The times of Apollonius
00:05:23 1.2 The short autobiography of Apollonius
00:10:32 2 Documented works of Apollonius
00:12:42 2.1 iConics/i
00:17:45 2.1.1 Book I
00:21:23 2.1.2 Book II
00:22:52 2.1.3 Book III
00:23:35 2.1.4 Book IV
00:24:44 2.1.5 Book V
00:25:21 2.1.6 Book VI
00:30:46 2.1.7 Book VII
00:33:41 2.2 Lost and reconstructed works described by Pappus
00:38:12 2.2.1 iDe Rationis Sectione/i
00:39:13 2.2.2 iDe Spatii Sectione/i
00:39:40 2.2.3 iDe Sectione Determinata/i
00:40:14 2.2.4 iDe Tactionibus/i
00:41:20 2.2.5 iDe Inclinationibus/i
00:42:21 2.2.6 iDe Locis Planis/i
00:42:52 2.3 Lost works mentioned by other ancient writers
00:43:30 2.4 Early p...
published: 04 Oct 2019
4:07
Apollonius of Perga
Apollonius was an influential Greek mathematician and astronomer, born in a region of what is now Turkey. He became known as the "Great Geometer."
My other You...
Apollonius was an influential Greek mathematician and astronomer, born in a region of what is now Turkey. He became known as the "Great Geometer."
My other YouTube channels:
The Science Fiction Rock Experience ( the music show I produce):
https://www.youtube.com/@ScienceFictionRockExperience
My science and music channel::
https://www.youtube.com/@drdaviddarling
Science World (with Emrah Polat):
https://www.youtube.com/@ScienceWorld1
My website: https://www.daviddarling.info
My latest book is available here: https://oneworld-publications.com/work/the-biggest-number-in-the-world/
https://wn.com/Apollonius_Of_Perga
Apollonius was an influential Greek mathematician and astronomer, born in a region of what is now Turkey. He became known as the "Great Geometer."
My other YouTube channels:
The Science Fiction Rock Experience ( the music show I produce):
https://www.youtube.com/@ScienceFictionRockExperience
My science and music channel::
https://www.youtube.com/@drdaviddarling
Science World (with Emrah Polat):
https://www.youtube.com/@ScienceWorld1
My website: https://www.daviddarling.info
My latest book is available here: https://oneworld-publications.com/work/the-biggest-number-in-the-world/
- published: 05 Oct 2022
- views: 3128
1:36
Apollonius of Perga
Dr Sergis discusses the contributions of another mathematician, Apollonius of Perga.
*************
The Dr Sergis Academy is a highly respected tutoring centr...
Dr Sergis discusses the contributions of another mathematician, Apollonius of Perga.
*************
The Dr Sergis Academy is a highly respected tutoring centre based in Enfield, North London. 🧡
For more information, please visit www.dsacademy.co.uk
We are also on Facebook, Instagram, TikTok and X, just search for the Dr Sergis Academy. 🔎
#enfield #thedrsergisacademy #mathematician #apolloniusofperga #tuition #tutors #ks2 #ks3 #gcse #alevel #english #maths #science
https://wn.com/Apollonius_Of_Perga
Dr Sergis discusses the contributions of another mathematician, Apollonius of Perga.
*************
The Dr Sergis Academy is a highly respected tutoring centre based in Enfield, North London. 🧡
For more information, please visit www.dsacademy.co.uk
We are also on Facebook, Instagram, TikTok and X, just search for the Dr Sergis Academy. 🔎
#enfield #thedrsergisacademy #mathematician #apolloniusofperga #tuition #tutors #ks2 #ks3 #gcse #alevel #english #maths #science
- published: 09 Dec 2023
- views: 26
1:20
Apollonius Cone
Apollonius of Perga was a Greek mathematician who lived from approximately 260 BC to 190 BC. One of his major written works, parts of which still exist, dealt w...
Apollonius of Perga was a Greek mathematician who lived from approximately 260 BC to 190 BC. One of his major written works, parts of which still exist, dealt with what is termed Conic Sections. His work is said to have influenced many later scholars, including Isaac Newton and René Descartes.
Conic Sections deals with the shapes that you get when a cone is intersected by a plane. Essentially you can get one of four shapes, a circle, an ellipse, a parabola or a hyperbola. It is believed to be Apollonius who gave the last three shapes the names we use today.
There was a time when most children in school learned about conic sections, although it is not so common nowadays.
This beautiful hand made wooden version of the Apollonius Cone stands 27 cm high. It comes apart into five pieces, and shows the four shapes that are created by the different sections through the cone.
It would make an elegant addition to any room, but will be especially appreciated by mathematicians.
http://www.grand-illusions.com/acatalog/Apollonius_Cone.html
https://wn.com/Apollonius_Cone
Apollonius of Perga was a Greek mathematician who lived from approximately 260 BC to 190 BC. One of his major written works, parts of which still exist, dealt with what is termed Conic Sections. His work is said to have influenced many later scholars, including Isaac Newton and René Descartes.
Conic Sections deals with the shapes that you get when a cone is intersected by a plane. Essentially you can get one of four shapes, a circle, an ellipse, a parabola or a hyperbola. It is believed to be Apollonius who gave the last three shapes the names we use today.
There was a time when most children in school learned about conic sections, although it is not so common nowadays.
This beautiful hand made wooden version of the Apollonius Cone stands 27 cm high. It comes apart into five pieces, and shows the four shapes that are created by the different sections through the cone.
It would make an elegant addition to any room, but will be especially appreciated by mathematicians.
http://www.grand-illusions.com/acatalog/Apollonius_Cone.html
- published: 01 Mar 2012
- views: 91347
43:49
Apollonius' circle construction problems | Famous Math Problems 3 | NJ Wildberger
Around 200 B.C., Apollonius of Perga, the greatest geometer of all time, gave a series of related problems; how to construct a circle in the plane touching thre...
Around 200 B.C., Apollonius of Perga, the greatest geometer of all time, gave a series of related problems; how to construct a circle in the plane touching three objects, where the objects are either a point (P), a line (L) and or a circle (C). Many mathematicians have studied this most famous of all geometric problems.
In this video we give some of the history, discuss Euclid's solution to two of the problems (which predate Apollonius' work), and summarize some circle geometry that is relevant to the subject: the Subtended spreads theorem from Rational Trigonometry, the power of a point wrt a circle, the pole-polar duality determined between points and lines by a circle, and inversion.
We also discuss the importance of constructions in both education and theoretical work---a point which has gradually been diminished over the years. Modern mathematics can learn a lot by re-thinking its attitude to the existence of mathematical objects by understanding the clarity that arises from the ancient Greek, and 17th and 18th century insistence on explicit constructions.
This problem has aspects which range in difficulty from 1 to 5. It is a great challenge for aspiring geometers.
************************
Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of the lectures for various Playlists: great for review, study and summary.
My research papers can be found at my Research Gate page, at https://www.researchgate.net/profile/Norman_Wildberger
My blog is at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things.
Online courses will be developed at openlearning.com. The first one, already underway is Algebraic Calculus One at https://www.openlearning.com/courses/algebraic-calculus-one/ Please join us for an exciting new approach to one of mathematics' most important subjects!
If you would like to support these new initiatives for mathematics education and research, please consider becoming a Patron of this Channel at https://www.patreon.com/njwildberger Your support would be much appreciated.
Here are the Insights into Mathematics Playlists:
https://www.youtube.com/playlist?list=PL55C7C83781CF4316
https://www.youtube.com/playlist?list=PL3C58498718451C47
https://www.youtube.com/playlist?list=PL5A714C94D40392AB
https://www.youtube.com/playlist?list=PLIljB45xT85BhzJ-oWNug1YtUjfWp1qAp
https://www.youtube.com/playlist?list=PLIljB45xT85Bfc-S4WHvTIM7E-ir3nAOf
https://www.youtube.com/playlist?list=PLIljB45xT85D94vHAB8joyFTH4dmVJ_Fw
https://www.youtube.com/playlist?list=PL8403C2F0C89B1333
https://www.youtube.com/playlist?list=PLIljB45xT85CcGpZpO542YLPeDIf1jqXK
https://www.youtube.com/playlist?list=PLIljB45xT85Aqe2b4FBWUGJdYROT6-o4e
https://www.youtube.com/playlist?list=PLIljB45xT85DB7CzoFWvA920NES3g8tJH
https://www.youtube.com/playlist?list=PLIljB45xT85A-qCypcmZqRvaS1pGXpTua
https://www.youtube.com/playlist?list=PLIljB45xT85DH__ZzGQWQrVRxlbKh-Nsa
https://www.youtube.com/playlist?list=PLIljB45xT85CdeBmQZ2QiCEnPQn5KQ6ov
https://www.youtube.com/playlist?list=PLIljB45xT85AMigTyprOuf__daeklnLse
https://www.youtube.com/playlist?list=PLIljB45xT85CnIGIWb7tH1F_S2PyOC8rb
https://www.youtube.com/playlist?list=PLIljB45xT85CN9oJ4gYkuSQQhAtpIucuI
https://www.youtube.com/playlist?list=PLIljB45xT85DWUiFYYGqJVtfnkUFWkKtP
https://www.youtube.com/playlist?list=PL6763F57A61FE6FE8
https://www.youtube.com/playlist?list=PLBF39AFBBC3FB30AF
https://www.youtube.com/playlist?list=PLIljB45xT85DSrlV6NX8RMBksZhdTHtwW
https://www.youtube.com/playlist?list=PLIljB45xT85Bmcc9ksBOAKgIZAl0BwPg7
Here are the Wild Egg Maths Playlists (some available only to Members!)
https://www.youtube.com/playlist?list=PLzdiPTrEWyz4rKFN541wFKvKPSg5Ea6XB
https://www.youtube.com/playlist?list=PLzdiPTrEWyz4VlOppC5CN0D0GjrjvBGKy
https://www.youtube.com/playlist?list=PLzdiPTrEWyz5VLVr-0LPPgm4T1mtU_DG-
https://www.youtube.com/playlist?list=PLzdiPTrEWyz6zpIZ4Y_RK9zyJ9OufqNJv
https://www.youtube.com/playlist?list=PLzdiPTrEWyz7hk_Kzj4zDF_kUXBCtiGn6
https://www.youtube.com/playlist?list=PLzdiPTrEWyz5j1BJdXBw1MFst_nQAqzZ_
https://www.youtube.com/playlist?list=PLzdiPTrEWyz5HWgaVkhIwpGVKi6fciRxW
https://www.youtube.com/playlist?list=PLzdiPTrEWyz5HBT_Yo1G4DfeqUfI9zkKM
https://www.youtube.com/playlist?list=PLzdiPTrEWyz7LKbuJAHaXAhRj2ylD0OoX
https://www.youtube.com/playlist?list=PLzdiPTrEWyz6VcJQ5xcuqY6g4DWjvpmjM
https://www.youtube.com/playlist?list=PLzdiPTrEWyz6MwUTOHRgC0oIxVtaHGQBd
https://www.youtube.com/playlist?list=PLzdiPTrEWyz4KD007Ge10dfrDVc4YwlYS
https://www.youtube.com/playlist?list=PLzdiPTrEWyz4IknbwXMEVwxOWS3z1Ch3C
************************
https://wn.com/Apollonius'_Circle_Construction_Problems_|_Famous_Math_Problems_3_|_Nj_Wildberger
Around 200 B.C., Apollonius of Perga, the greatest geometer of all time, gave a series of related problems; how to construct a circle in the plane touching three objects, where the objects are either a point (P), a line (L) and or a circle (C). Many mathematicians have studied this most famous of all geometric problems.
In this video we give some of the history, discuss Euclid's solution to two of the problems (which predate Apollonius' work), and summarize some circle geometry that is relevant to the subject: the Subtended spreads theorem from Rational Trigonometry, the power of a point wrt a circle, the pole-polar duality determined between points and lines by a circle, and inversion.
We also discuss the importance of constructions in both education and theoretical work---a point which has gradually been diminished over the years. Modern mathematics can learn a lot by re-thinking its attitude to the existence of mathematical objects by understanding the clarity that arises from the ancient Greek, and 17th and 18th century insistence on explicit constructions.
This problem has aspects which range in difficulty from 1 to 5. It is a great challenge for aspiring geometers.
************************
Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of the lectures for various Playlists: great for review, study and summary.
My research papers can be found at my Research Gate page, at https://www.researchgate.net/profile/Norman_Wildberger
My blog is at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things.
Online courses will be developed at openlearning.com. The first one, already underway is Algebraic Calculus One at https://www.openlearning.com/courses/algebraic-calculus-one/ Please join us for an exciting new approach to one of mathematics' most important subjects!
If you would like to support these new initiatives for mathematics education and research, please consider becoming a Patron of this Channel at https://www.patreon.com/njwildberger Your support would be much appreciated.
Here are the Insights into Mathematics Playlists:
https://www.youtube.com/playlist?list=PL55C7C83781CF4316
https://www.youtube.com/playlist?list=PL3C58498718451C47
https://www.youtube.com/playlist?list=PL5A714C94D40392AB
https://www.youtube.com/playlist?list=PLIljB45xT85BhzJ-oWNug1YtUjfWp1qAp
https://www.youtube.com/playlist?list=PLIljB45xT85Bfc-S4WHvTIM7E-ir3nAOf
https://www.youtube.com/playlist?list=PLIljB45xT85D94vHAB8joyFTH4dmVJ_Fw
https://www.youtube.com/playlist?list=PL8403C2F0C89B1333
https://www.youtube.com/playlist?list=PLIljB45xT85CcGpZpO542YLPeDIf1jqXK
https://www.youtube.com/playlist?list=PLIljB45xT85Aqe2b4FBWUGJdYROT6-o4e
https://www.youtube.com/playlist?list=PLIljB45xT85DB7CzoFWvA920NES3g8tJH
https://www.youtube.com/playlist?list=PLIljB45xT85A-qCypcmZqRvaS1pGXpTua
https://www.youtube.com/playlist?list=PLIljB45xT85DH__ZzGQWQrVRxlbKh-Nsa
https://www.youtube.com/playlist?list=PLIljB45xT85CdeBmQZ2QiCEnPQn5KQ6ov
https://www.youtube.com/playlist?list=PLIljB45xT85AMigTyprOuf__daeklnLse
https://www.youtube.com/playlist?list=PLIljB45xT85CnIGIWb7tH1F_S2PyOC8rb
https://www.youtube.com/playlist?list=PLIljB45xT85CN9oJ4gYkuSQQhAtpIucuI
https://www.youtube.com/playlist?list=PLIljB45xT85DWUiFYYGqJVtfnkUFWkKtP
https://www.youtube.com/playlist?list=PL6763F57A61FE6FE8
https://www.youtube.com/playlist?list=PLBF39AFBBC3FB30AF
https://www.youtube.com/playlist?list=PLIljB45xT85DSrlV6NX8RMBksZhdTHtwW
https://www.youtube.com/playlist?list=PLIljB45xT85Bmcc9ksBOAKgIZAl0BwPg7
Here are the Wild Egg Maths Playlists (some available only to Members!)
https://www.youtube.com/playlist?list=PLzdiPTrEWyz4rKFN541wFKvKPSg5Ea6XB
https://www.youtube.com/playlist?list=PLzdiPTrEWyz4VlOppC5CN0D0GjrjvBGKy
https://www.youtube.com/playlist?list=PLzdiPTrEWyz5VLVr-0LPPgm4T1mtU_DG-
https://www.youtube.com/playlist?list=PLzdiPTrEWyz6zpIZ4Y_RK9zyJ9OufqNJv
https://www.youtube.com/playlist?list=PLzdiPTrEWyz7hk_Kzj4zDF_kUXBCtiGn6
https://www.youtube.com/playlist?list=PLzdiPTrEWyz5j1BJdXBw1MFst_nQAqzZ_
https://www.youtube.com/playlist?list=PLzdiPTrEWyz5HWgaVkhIwpGVKi6fciRxW
https://www.youtube.com/playlist?list=PLzdiPTrEWyz5HBT_Yo1G4DfeqUfI9zkKM
https://www.youtube.com/playlist?list=PLzdiPTrEWyz7LKbuJAHaXAhRj2ylD0OoX
https://www.youtube.com/playlist?list=PLzdiPTrEWyz6VcJQ5xcuqY6g4DWjvpmjM
https://www.youtube.com/playlist?list=PLzdiPTrEWyz6MwUTOHRgC0oIxVtaHGQBd
https://www.youtube.com/playlist?list=PLzdiPTrEWyz4KD007Ge10dfrDVc4YwlYS
https://www.youtube.com/playlist?list=PLzdiPTrEWyz4IknbwXMEVwxOWS3z1Ch3C
************************
- published: 23 Jan 2013
- views: 30355
11:47
Problem of Apollonius - what does it teach us about problem solving?
This video uses the problem of Apollonius as a way to introduce circle inversion and an important problem-solving technique - transforming a hard problem into a...
This video uses the problem of Apollonius as a way to introduce circle inversion and an important problem-solving technique - transforming a hard problem into a simpler one; then solve for the simpler, transformed version of the problem before doing the inverse transformation so that we obtain the solution to the hard problem. This problem-solving technique is the motive behind Fourier transform and other transforms like that (Laplace / Mellin): transforming the forced ODEs to a polynomial function, which is much easier than the original problem.
Circle inversion is also an important technique in Euclidean geometry - it grants access to many more advanced geometric problems, like Pappus chain. This is because it really changes the form of the geometric object as opposed to translation, reflection, rotation and dilations; but it still preserves circles or lines (or generalised circles - lines can be thought of as circles with infinite radius), i.e. generalised circles mapped to generalised circles, and its anticonformal property: preserving angles while reversing orientations. This can be used to prove that complex inversion 1/z is conformal; and hence an important tool in complex analysis as well, not just Euclidean geometry.
Most people know other solutions to the problem of Apollonius, but I think this is a much easier solution to understand. If you are interested in checking out other solutions, be prepared to hold on to all your knowledge of Euclidean geometry (like power of a point / radical axis and so on) and check out these links:
(1) Problem of Apollonius wikipedia page: https://en.wikipedia.org/wiki/Problem_of_Apollonius
(2) Inversive solution of Problem of Apollonius: https://www.ntg.nl/maps/40/03.pdf
(3) Other methods of solving: https://www.cut-the-knot.org/pythagoras/Apollonius.shtml
http://users.math.uoc.gr/~pamfilos/eGallery/problems/ApolloniusProblem.pdf
Source of photo of Apollonius: https://www.researchgate.net/figure/Apollonius-of-Perga-252-170-BC_fig1_317673086
Correction:
At around 8:12, I should have said "we already know that *the centre of* the solution circle...", but it shouldn't materially impact the video. It's just a bad writing of script on my part, which I apologise.
Other than commenting on the video, you are very welcome to fill in a Google form linked below, which helps me make better videos by catering for your math levels:
https://forms.gle/QJ29hocF9uQAyZyH6
If you want to know more interesting Mathematics, stay tuned for the next video!
SUBSCRIBE and see you in the next video!
If you are wondering how I made all these videos, even though it is stylistically similar to 3Blue1Brown, I don't use his animation engine Manim, but I will probably reveal how I did it in a potential subscriber milestone, so do subscribe!
Social media:
Facebook: https://www.facebook.com/mathemaniacyt
Instagram: https://www.instagram.com/_mathemaniac_/
Twitter: https://twitter.com/mathemaniacyt
For my contact email, check my About page on a PC.
See you next time!
https://wn.com/Problem_Of_Apollonius_What_Does_It_Teach_US_About_Problem_Solving
This video uses the problem of Apollonius as a way to introduce circle inversion and an important problem-solving technique - transforming a hard problem into a simpler one; then solve for the simpler, transformed version of the problem before doing the inverse transformation so that we obtain the solution to the hard problem. This problem-solving technique is the motive behind Fourier transform and other transforms like that (Laplace / Mellin): transforming the forced ODEs to a polynomial function, which is much easier than the original problem.
Circle inversion is also an important technique in Euclidean geometry - it grants access to many more advanced geometric problems, like Pappus chain. This is because it really changes the form of the geometric object as opposed to translation, reflection, rotation and dilations; but it still preserves circles or lines (or generalised circles - lines can be thought of as circles with infinite radius), i.e. generalised circles mapped to generalised circles, and its anticonformal property: preserving angles while reversing orientations. This can be used to prove that complex inversion 1/z is conformal; and hence an important tool in complex analysis as well, not just Euclidean geometry.
Most people know other solutions to the problem of Apollonius, but I think this is a much easier solution to understand. If you are interested in checking out other solutions, be prepared to hold on to all your knowledge of Euclidean geometry (like power of a point / radical axis and so on) and check out these links:
(1) Problem of Apollonius wikipedia page: https://en.wikipedia.org/wiki/Problem_of_Apollonius
(2) Inversive solution of Problem of Apollonius: https://www.ntg.nl/maps/40/03.pdf
(3) Other methods of solving: https://www.cut-the-knot.org/pythagoras/Apollonius.shtml
http://users.math.uoc.gr/~pamfilos/eGallery/problems/ApolloniusProblem.pdf
Source of photo of Apollonius: https://www.researchgate.net/figure/Apollonius-of-Perga-252-170-BC_fig1_317673086
Correction:
At around 8:12, I should have said "we already know that *the centre of* the solution circle...", but it shouldn't materially impact the video. It's just a bad writing of script on my part, which I apologise.
Other than commenting on the video, you are very welcome to fill in a Google form linked below, which helps me make better videos by catering for your math levels:
https://forms.gle/QJ29hocF9uQAyZyH6
If you want to know more interesting Mathematics, stay tuned for the next video!
SUBSCRIBE and see you in the next video!
If you are wondering how I made all these videos, even though it is stylistically similar to 3Blue1Brown, I don't use his animation engine Manim, but I will probably reveal how I did it in a potential subscriber milestone, so do subscribe!
Social media:
Facebook: https://www.facebook.com/mathemaniacyt
Instagram: https://www.instagram.com/_mathemaniac_/
Twitter: https://twitter.com/mathemaniacyt
For my contact email, check my About page on a PC.
See you next time!
- published: 21 Sep 2020
- views: 52329
40:38
Apollonius and polarity | Universal Hyperbolic Geometry 1 | NJ Wildberger
This is the start of a new course on hyperbolic geometry that features a revolutionary simplifed approach to the subject, framing it in terms of classical proje...
This is the start of a new course on hyperbolic geometry that features a revolutionary simplifed approach to the subject, framing it in terms of classical projective geometry and the study of a distinguished circle. This subject will be called Universal Hyperbolic Geometry, as it extends the subject to arbitrary fields, as well as to the outside of the light cone/null circle.
We begin by going back to Apollonius of Perga (in present day Turkey, not Italy!) and his understanding of the crucial role of polarity in studying conics, in particular the circle. Given a fixed circle, to each point in the plane we associate a line called the polar, and conversely to a line we associated a point called its pole. This duality is all important for hyperbolic geometry
CONTENT SUMMARY: conics @00:00 the circle @5:00 Thales thm @05:20 Greek m'ment @06:20 polarity, pole of A and polar of a @09:06 def' of polarity (projective def') @14:40 Polar independence Theorem @19:13 projective def' of polarity @23:34 polar of a point inside the circle @25:10 3-way symetry @26:40 hands_on experience @27:52 pole, polar starting with four points on circle @29:12 Quadrangle and quadralateral @29:26 Polar duality thm @31:30 The distinguished circle @37:40 Pole of a line thm @38:00 (THANKS to EmptySpaceEnterprise)
************************
Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of the lectures for various Playlists: great for review, study and summary.
My research papers can be found at my Research Gate page, at https://www.researchgate.net/profile/Norman_Wildberger
My blog is at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things.
Online courses will be developed at openlearning.com. The first one, already underway is Algebraic Calculus One at https://www.openlearning.com/courses/algebraic-calculus-one/ Please join us for an exciting new approach to one of mathematics' most important subjects!
If you would like to support these new initiatives for mathematics education and research, please consider becoming a Patron of this Channel at https://www.patreon.com/njwildberger Your support would be much appreciated.
Here are the Insights into Mathematics Playlists:
https://www.youtube.com/playlist?list=PL55C7C83781CF4316
https://www.youtube.com/playlist?list=PL3C58498718451C47
https://www.youtube.com/playlist?list=PL5A714C94D40392AB
https://www.youtube.com/playlist?list=PLIljB45xT85BhzJ-oWNug1YtUjfWp1qAp
https://www.youtube.com/playlist?list=PLIljB45xT85Bfc-S4WHvTIM7E-ir3nAOf
https://www.youtube.com/playlist?list=PLIljB45xT85D94vHAB8joyFTH4dmVJ_Fw
https://www.youtube.com/playlist?list=PL8403C2F0C89B1333
https://www.youtube.com/playlist?list=PLIljB45xT85CcGpZpO542YLPeDIf1jqXK
https://www.youtube.com/playlist?list=PLIljB45xT85Aqe2b4FBWUGJdYROT6-o4e
https://www.youtube.com/playlist?list=PLIljB45xT85DB7CzoFWvA920NES3g8tJH
https://www.youtube.com/playlist?list=PLIljB45xT85A-qCypcmZqRvaS1pGXpTua
https://www.youtube.com/playlist?list=PLIljB45xT85DH__ZzGQWQrVRxlbKh-Nsa
https://www.youtube.com/playlist?list=PLIljB45xT85CdeBmQZ2QiCEnPQn5KQ6ov
https://www.youtube.com/playlist?list=PLIljB45xT85AMigTyprOuf__daeklnLse
https://www.youtube.com/playlist?list=PLIljB45xT85CnIGIWb7tH1F_S2PyOC8rb
https://www.youtube.com/playlist?list=PLIljB45xT85CN9oJ4gYkuSQQhAtpIucuI
https://www.youtube.com/playlist?list=PLIljB45xT85DWUiFYYGqJVtfnkUFWkKtP
https://www.youtube.com/playlist?list=PL6763F57A61FE6FE8
https://www.youtube.com/playlist?list=PLBF39AFBBC3FB30AF
https://www.youtube.com/playlist?list=PLIljB45xT85DSrlV6NX8RMBksZhdTHtwW
https://www.youtube.com/playlist?list=PLIljB45xT85Bmcc9ksBOAKgIZAl0BwPg7
Here are the Wild Egg Maths Playlists (some available only to Members!)
https://www.youtube.com/playlist?list=PLzdiPTrEWyz4rKFN541wFKvKPSg5Ea6XB
https://www.youtube.com/playlist?list=PLzdiPTrEWyz4VlOppC5CN0D0GjrjvBGKy
https://www.youtube.com/playlist?list=PLzdiPTrEWyz5VLVr-0LPPgm4T1mtU_DG-
https://www.youtube.com/playlist?list=PLzdiPTrEWyz6zpIZ4Y_RK9zyJ9OufqNJv
https://www.youtube.com/playlist?list=PLzdiPTrEWyz7hk_Kzj4zDF_kUXBCtiGn6
https://www.youtube.com/playlist?list=PLzdiPTrEWyz5j1BJdXBw1MFst_nQAqzZ_
https://www.youtube.com/playlist?list=PLzdiPTrEWyz5HWgaVkhIwpGVKi6fciRxW
https://www.youtube.com/playlist?list=PLzdiPTrEWyz5HBT_Yo1G4DfeqUfI9zkKM
https://www.youtube.com/playlist?list=PLzdiPTrEWyz7LKbuJAHaXAhRj2ylD0OoX
https://www.youtube.com/playlist?list=PLzdiPTrEWyz6VcJQ5xcuqY6g4DWjvpmjM
https://www.youtube.com/playlist?list=PLzdiPTrEWyz6MwUTOHRgC0oIxVtaHGQBd
https://www.youtube.com/playlist?list=PLzdiPTrEWyz4KD007Ge10dfrDVc4YwlYS
https://www.youtube.com/playlist?list=PLzdiPTrEWyz4IknbwXMEVwxOWS3z1Ch3C
************************
https://wn.com/Apollonius_And_Polarity_|_Universal_Hyperbolic_Geometry_1_|_Nj_Wildberger
This is the start of a new course on hyperbolic geometry that features a revolutionary simplifed approach to the subject, framing it in terms of classical projective geometry and the study of a distinguished circle. This subject will be called Universal Hyperbolic Geometry, as it extends the subject to arbitrary fields, as well as to the outside of the light cone/null circle.
We begin by going back to Apollonius of Perga (in present day Turkey, not Italy!) and his understanding of the crucial role of polarity in studying conics, in particular the circle. Given a fixed circle, to each point in the plane we associate a line called the polar, and conversely to a line we associated a point called its pole. This duality is all important for hyperbolic geometry
CONTENT SUMMARY: conics @00:00 the circle @5:00 Thales thm @05:20 Greek m'ment @06:20 polarity, pole of A and polar of a @09:06 def' of polarity (projective def') @14:40 Polar independence Theorem @19:13 projective def' of polarity @23:34 polar of a point inside the circle @25:10 3-way symetry @26:40 hands_on experience @27:52 pole, polar starting with four points on circle @29:12 Quadrangle and quadralateral @29:26 Polar duality thm @31:30 The distinguished circle @37:40 Pole of a line thm @38:00 (THANKS to EmptySpaceEnterprise)
************************
Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of the lectures for various Playlists: great for review, study and summary.
My research papers can be found at my Research Gate page, at https://www.researchgate.net/profile/Norman_Wildberger
My blog is at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things.
Online courses will be developed at openlearning.com. The first one, already underway is Algebraic Calculus One at https://www.openlearning.com/courses/algebraic-calculus-one/ Please join us for an exciting new approach to one of mathematics' most important subjects!
If you would like to support these new initiatives for mathematics education and research, please consider becoming a Patron of this Channel at https://www.patreon.com/njwildberger Your support would be much appreciated.
Here are the Insights into Mathematics Playlists:
https://www.youtube.com/playlist?list=PL55C7C83781CF4316
https://www.youtube.com/playlist?list=PL3C58498718451C47
https://www.youtube.com/playlist?list=PL5A714C94D40392AB
https://www.youtube.com/playlist?list=PLIljB45xT85BhzJ-oWNug1YtUjfWp1qAp
https://www.youtube.com/playlist?list=PLIljB45xT85Bfc-S4WHvTIM7E-ir3nAOf
https://www.youtube.com/playlist?list=PLIljB45xT85D94vHAB8joyFTH4dmVJ_Fw
https://www.youtube.com/playlist?list=PL8403C2F0C89B1333
https://www.youtube.com/playlist?list=PLIljB45xT85CcGpZpO542YLPeDIf1jqXK
https://www.youtube.com/playlist?list=PLIljB45xT85Aqe2b4FBWUGJdYROT6-o4e
https://www.youtube.com/playlist?list=PLIljB45xT85DB7CzoFWvA920NES3g8tJH
https://www.youtube.com/playlist?list=PLIljB45xT85A-qCypcmZqRvaS1pGXpTua
https://www.youtube.com/playlist?list=PLIljB45xT85DH__ZzGQWQrVRxlbKh-Nsa
https://www.youtube.com/playlist?list=PLIljB45xT85CdeBmQZ2QiCEnPQn5KQ6ov
https://www.youtube.com/playlist?list=PLIljB45xT85AMigTyprOuf__daeklnLse
https://www.youtube.com/playlist?list=PLIljB45xT85CnIGIWb7tH1F_S2PyOC8rb
https://www.youtube.com/playlist?list=PLIljB45xT85CN9oJ4gYkuSQQhAtpIucuI
https://www.youtube.com/playlist?list=PLIljB45xT85DWUiFYYGqJVtfnkUFWkKtP
https://www.youtube.com/playlist?list=PL6763F57A61FE6FE8
https://www.youtube.com/playlist?list=PLBF39AFBBC3FB30AF
https://www.youtube.com/playlist?list=PLIljB45xT85DSrlV6NX8RMBksZhdTHtwW
https://www.youtube.com/playlist?list=PLIljB45xT85Bmcc9ksBOAKgIZAl0BwPg7
Here are the Wild Egg Maths Playlists (some available only to Members!)
https://www.youtube.com/playlist?list=PLzdiPTrEWyz4rKFN541wFKvKPSg5Ea6XB
https://www.youtube.com/playlist?list=PLzdiPTrEWyz4VlOppC5CN0D0GjrjvBGKy
https://www.youtube.com/playlist?list=PLzdiPTrEWyz5VLVr-0LPPgm4T1mtU_DG-
https://www.youtube.com/playlist?list=PLzdiPTrEWyz6zpIZ4Y_RK9zyJ9OufqNJv
https://www.youtube.com/playlist?list=PLzdiPTrEWyz7hk_Kzj4zDF_kUXBCtiGn6
https://www.youtube.com/playlist?list=PLzdiPTrEWyz5j1BJdXBw1MFst_nQAqzZ_
https://www.youtube.com/playlist?list=PLzdiPTrEWyz5HWgaVkhIwpGVKi6fciRxW
https://www.youtube.com/playlist?list=PLzdiPTrEWyz5HBT_Yo1G4DfeqUfI9zkKM
https://www.youtube.com/playlist?list=PLzdiPTrEWyz7LKbuJAHaXAhRj2ylD0OoX
https://www.youtube.com/playlist?list=PLzdiPTrEWyz6VcJQ5xcuqY6g4DWjvpmjM
https://www.youtube.com/playlist?list=PLzdiPTrEWyz6MwUTOHRgC0oIxVtaHGQBd
https://www.youtube.com/playlist?list=PLzdiPTrEWyz4KD007Ge10dfrDVc4YwlYS
https://www.youtube.com/playlist?list=PLzdiPTrEWyz4IknbwXMEVwxOWS3z1Ch3C
************************
- published: 18 Apr 2011
- views: 53377
1:00
Apollonius of Perga | Heroes of Mathematics | Mathematicians
Discover the profound mathematical contributions of Apollonius of Perga in this enlightening video. Apollonius, a mathematical luminary from the 3rd century BCE...
Discover the profound mathematical contributions of Apollonius of Perga in this enlightening video. Apollonius, a mathematical luminary from the 3rd century BCE, revolutionized the study of conic sections, encompassing circles, ellipses, parabolas, and hyperbolas. His groundbreaking theorems and concepts continue to shape modern mathematics and various scientific fields. Join us as we delve into Apollonius's life and his enduring legacy, offering a glimpse into the ancient world of mathematics. Whether you're a math enthusiast, history buff, or simply curious, this video provides a captivating journey through the foundations of geometry. Don't miss out – like, subscribe, and share to spread the brilliance of Apollonius of Perga and his timeless contributions to mathematics and science.
Thanks for watching! If you found these tips helpful, please like and subscribe to our channel. Share your reviews in the comments below!
#short #shorts #shortvideo #viralshort #viral #viralshorts #shortsfeed #shortszone #shortsidea #shortsideas #history #historyfacts #historygk #historyofmath #mathematician #mathematicians #mathematicianhistory #mathhub #mathhistory #numeric #hub #numerichub #funfact #factshorts #factshorts #mathfacts #mathisfun #mathisawesome #mathiscool #mathislife #conics #apollonius #perga #apolloniancircle #gasket #lunar #crater #lunarcrater #apolloniancrater #mathzone
https://wn.com/Apollonius_Of_Perga_|_Heroes_Of_Mathematics_|_Mathematicians
Discover the profound mathematical contributions of Apollonius of Perga in this enlightening video. Apollonius, a mathematical luminary from the 3rd century BCE, revolutionized the study of conic sections, encompassing circles, ellipses, parabolas, and hyperbolas. His groundbreaking theorems and concepts continue to shape modern mathematics and various scientific fields. Join us as we delve into Apollonius's life and his enduring legacy, offering a glimpse into the ancient world of mathematics. Whether you're a math enthusiast, history buff, or simply curious, this video provides a captivating journey through the foundations of geometry. Don't miss out – like, subscribe, and share to spread the brilliance of Apollonius of Perga and his timeless contributions to mathematics and science.
Thanks for watching! If you found these tips helpful, please like and subscribe to our channel. Share your reviews in the comments below!
#short #shorts #shortvideo #viralshort #viral #viralshorts #shortsfeed #shortszone #shortsidea #shortsideas #history #historyfacts #historygk #historyofmath #mathematician #mathematicians #mathematicianhistory #mathhub #mathhistory #numeric #hub #numerichub #funfact #factshorts #factshorts #mathfacts #mathisfun #mathisawesome #mathiscool #mathislife #conics #apollonius #perga #apolloniancircle #gasket #lunar #crater #lunarcrater #apolloniancrater #mathzone
- published: 13 Sep 2023
- views: 419
1:03:17
Apollonius of Perga | Wikipedia audio article
This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Apollonius_of_Perga
00:00:50 1 Life
00:04:34 1.1 The times of Apollonius
...
This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Apollonius_of_Perga
00:00:50 1 Life
00:04:34 1.1 The times of Apollonius
00:05:23 1.2 The short autobiography of Apollonius
00:10:32 2 Documented works of Apollonius
00:12:42 2.1 iConics/i
00:17:45 2.1.1 Book I
00:21:23 2.1.2 Book II
00:22:52 2.1.3 Book III
00:23:35 2.1.4 Book IV
00:24:44 2.1.5 Book V
00:25:21 2.1.6 Book VI
00:30:46 2.1.7 Book VII
00:33:41 2.2 Lost and reconstructed works described by Pappus
00:38:12 2.2.1 iDe Rationis Sectione/i
00:39:13 2.2.2 iDe Spatii Sectione/i
00:39:40 2.2.3 iDe Sectione Determinata/i
00:40:14 2.2.4 iDe Tactionibus/i
00:41:20 2.2.5 iDe Inclinationibus/i
00:42:21 2.2.6 iDe Locis Planis/i
00:42:52 2.3 Lost works mentioned by other ancient writers
00:43:30 2.4 Early printed editions
00:44:58 3 Ideas attributed to Apollonius by other writers
00:50:18 3.1 Apollonius' contribution to astronomy
00:50:30 3.2 Methods of Apollonius
00:50:55 3.2.1 Geometrical algebra
00:52:03 3.2.2 The coordinate system of Apollonius
00:54:59 3.2.3 The theory of proportions
00:58:56 4 Honors accorded by history
01:01:21 5 See also
01:02:41 6 Notes
01:02:44 7 References
01:02:55 8 External links
01:03:02 See also
01:03:12
Listening is a more natural way of learning, when compared to reading. Written language only began at around 3200 BC, but spoken language has existed long ago.
Learning by listening is a great way to:
- increases imagination and understanding
- improves your listening skills
- improves your own spoken accent
- learn while on the move
- reduce eye strain
Now learn the vast amount of general knowledge available on Wikipedia through audio (audio article). You could even learn subconsciously by playing the audio while you are sleeping! If you are planning to listen a lot, you could try using a bone conduction headphone, or a standard speaker instead of an earphone.
Listen on Google Assistant through Extra Audio:
https://assistant.google.com/services/invoke/uid/0000001a130b3f91
Other Wikipedia audio articles at:
https://www.youtube.com/results?search_query=wikipedia+tts
Upload your own Wikipedia articles through:
https://github.com/nodef/wikipedia-tts
Speaking Rate: 0.9888088068772672
Voice name: en-US-Wavenet-E
"I cannot teach anybody anything, I can only make them think."
- Socrates
SUMMARY
=======
Apollonius of Perga (Greek: Ἀπολλώνιος ὁ Περγαῖος; Latin: Apollonius Pergaeus; late 3rd – early 2nd centuries BC) was a Greek geometer and astronomer known for his theories on the topic of conic sections. Beginning from the theories of Euclid and Archimedes on the topic, he brought them to the state they were in just before the invention of analytic geometry. His definitions of the terms ellipse, parabola, and hyperbola are the ones in use today.
Apollonius worked on many other topics, including astronomy. Most of the work has not survived except in fragmentary references in other authors. His hypothesis of eccentric orbits to explain the apparently aberrant motion of the planets, commonly believed until the Middle Ages, was superseded during the Renaissance.
https://wn.com/Apollonius_Of_Perga_|_Wikipedia_Audio_Article
This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Apollonius_of_Perga
00:00:50 1 Life
00:04:34 1.1 The times of Apollonius
00:05:23 1.2 The short autobiography of Apollonius
00:10:32 2 Documented works of Apollonius
00:12:42 2.1 iConics/i
00:17:45 2.1.1 Book I
00:21:23 2.1.2 Book II
00:22:52 2.1.3 Book III
00:23:35 2.1.4 Book IV
00:24:44 2.1.5 Book V
00:25:21 2.1.6 Book VI
00:30:46 2.1.7 Book VII
00:33:41 2.2 Lost and reconstructed works described by Pappus
00:38:12 2.2.1 iDe Rationis Sectione/i
00:39:13 2.2.2 iDe Spatii Sectione/i
00:39:40 2.2.3 iDe Sectione Determinata/i
00:40:14 2.2.4 iDe Tactionibus/i
00:41:20 2.2.5 iDe Inclinationibus/i
00:42:21 2.2.6 iDe Locis Planis/i
00:42:52 2.3 Lost works mentioned by other ancient writers
00:43:30 2.4 Early printed editions
00:44:58 3 Ideas attributed to Apollonius by other writers
00:50:18 3.1 Apollonius' contribution to astronomy
00:50:30 3.2 Methods of Apollonius
00:50:55 3.2.1 Geometrical algebra
00:52:03 3.2.2 The coordinate system of Apollonius
00:54:59 3.2.3 The theory of proportions
00:58:56 4 Honors accorded by history
01:01:21 5 See also
01:02:41 6 Notes
01:02:44 7 References
01:02:55 8 External links
01:03:02 See also
01:03:12
Listening is a more natural way of learning, when compared to reading. Written language only began at around 3200 BC, but spoken language has existed long ago.
Learning by listening is a great way to:
- increases imagination and understanding
- improves your listening skills
- improves your own spoken accent
- learn while on the move
- reduce eye strain
Now learn the vast amount of general knowledge available on Wikipedia through audio (audio article). You could even learn subconsciously by playing the audio while you are sleeping! If you are planning to listen a lot, you could try using a bone conduction headphone, or a standard speaker instead of an earphone.
Listen on Google Assistant through Extra Audio:
https://assistant.google.com/services/invoke/uid/0000001a130b3f91
Other Wikipedia audio articles at:
https://www.youtube.com/results?search_query=wikipedia+tts
Upload your own Wikipedia articles through:
https://github.com/nodef/wikipedia-tts
Speaking Rate: 0.9888088068772672
Voice name: en-US-Wavenet-E
"I cannot teach anybody anything, I can only make them think."
- Socrates
SUMMARY
=======
Apollonius of Perga (Greek: Ἀπολλώνιος ὁ Περγαῖος; Latin: Apollonius Pergaeus; late 3rd – early 2nd centuries BC) was a Greek geometer and astronomer known for his theories on the topic of conic sections. Beginning from the theories of Euclid and Archimedes on the topic, he brought them to the state they were in just before the invention of analytic geometry. His definitions of the terms ellipse, parabola, and hyperbola are the ones in use today.
Apollonius worked on many other topics, including astronomy. Most of the work has not survived except in fragmentary references in other authors. His hypothesis of eccentric orbits to explain the apparently aberrant motion of the planets, commonly believed until the Middle Ages, was superseded during the Renaissance.
- published: 04 Oct 2019
- views: 107