A visual proof to a really beautiful result from projective geometry (this video replaces my previous upload `Projective Geometry 4: A Simple Proof to Desargues' Theorem', which had some issues)
@artr0x934 жыл бұрын
Thanks for this video, nicely visualized! I had a hard time with understanding this from my textbook but seeing the lines move around makes the argument much more clear
@DrDanielHoward4 ай бұрын
Excellent instruction.
@user-nr5tp2jo3u6 ай бұрын
It is famous projective theorem like Pappus's theorem, Brianchon's theorem and Pascal's theorem.
@pragalbhawasthi16183 жыл бұрын
So satisfying!! I am in love with all these videos.
@floretionguru29774 жыл бұрын
Wonderfully visualized and understood- thanks!
@MichaelMarteens2 жыл бұрын
Great video! Thank you so much for explaining so well.
@minhokim82636 жыл бұрын
It's just beautiful. Thank you for sharing this with us, I appreciate it.
@RichardSouthwell6 жыл бұрын
Minho Kim good of you to say, thank you
@dreamscorner985 жыл бұрын
It took you 2 years to understand this....😅😂
@fornlike4 ай бұрын
Thank you for this beautiful sharing. However, I have one question regarding this geometry. If it's not part of Euclidean geometries, how could it have been founded when the problem of Euclid's 5th postulate had not yet been resolved (By Gauss)?
@mailoisback3 жыл бұрын
What software did you use for drawing? Thank you!
@RichardSouthwell3 жыл бұрын
geogebra
@SimchaWaldman6 жыл бұрын
The first half of the proof here (if you indeed use the "projectivity principal" you only need half) is similar and easier to read and watch: proofwiki.org/wiki/Desargues%27_Theorem
@3kbschannel2888 жыл бұрын
Hi Richard, I understand why the lines meet when they diverge, when I think of it as if we are on a spherical round object, like the Earth. However, I can never understand why they do meet when they are parallel to each other! Because they never go further or closer to each other, and distance between them will stay like that infinitely! I think!
@clickaccept6 жыл бұрын
The notion of distance is not a concept of projective geometry. Projective geometry is an axiomatic system for studying points and lines. The axiom that every pair of lines meet at a point seems un-physical initially. However, one can obtain affine geometry from projective one (projective one is more fundamental). In a generic projective plane, single out one special line. Any pair of lines that meet that special line at the same point, we call "parallel". (there is one parallel "class" for each point on the special line). So, it starts out just as a labelling: You identify each parallel class with a point, and the set of all of them, with a line. But then magic happens, the new "points" and "line" satisfy exactly the same axioms and theorems as the rest of points and lines of the projective plane. So there is a kind of miracle of nature, that what we perceive as the infinity of a physical plane, behaves exactly like a line. And it is beautiful and powerful to treat it as such in mathematics.
@AbsolutelyRami2 жыл бұрын
@@clickaccept thats why in the 18th century, desargues’ theory was rejected, it was not realistic. “The noticeable return to the phenomena implicit in the method of physics and natural history during the eighteenth century, reinforced the status of Euclidean spatiality” Perez-Gomez (Architecture and the Crisis of Modern Science 1983)
@infernocow226 жыл бұрын
thanks James Corden
@zainabsalah48157 жыл бұрын
hey can u help me ? can u prove this theorem : if there are exactly "n" of points in line on " finite projective plane " the plane contains exactly " n^2 - n+1 " of points can u prove it please ?
@clickaccept6 жыл бұрын
1. Every line meets every other line at a point of the space. 2. If each line is incident with n points, then each point is incident with n lines, by the projective duality. 3. By (1) and (2), a given line is therefore incident with n*(n-1) other lines, which exhaust all lines of the space. 4. Add to this number, the original line, we have k=n^2-n+1 lines in total. 5. Total number of points coincides with total number of lines, again using the duality. This proof relies on the projective duality (and implicitly on transitivity of projective transformations).
@minhokim82638 жыл бұрын
The explanation isn't clear and still don't understand what he tried to tell on the grey triangle. Richard, you can do it better, don't you?
@duckymomo79354 жыл бұрын
Personally I feel like there’s too many lines to begin with Ie I feel, I could be wrong on pedagogy, like maybe drawing them as we go along might help
@myName-dg2qm9 жыл бұрын
Found it! :-)
@myName-dg2qm9 жыл бұрын
You: ... the intersection of planes. Me: Ohhhhhhhhhhhhhh........
@myName-dg2qm9 жыл бұрын
The lines at the corresponding sides of the triangles intersect, they are on intersecting planes, therefore the intesecting lines must be on a line (the intersection of the planes).
@myName-dg2qm9 жыл бұрын
This seems to imply that the three projected rays along the corresponding vertices of any two triangles in three dimentional space will intersect at a point. Wonder how to prove it.
@myName-dg2qm9 жыл бұрын
No, that ladder statement in incorrect! Ill still have to think around this a bit.
@RichardSouthwell9 жыл бұрын
myName haha, well im glad it helps, maybe you might find this book cover picture enjoyable