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I do math
I do math
= episode 20 experimental markup dump =
== Experimental Markup for player ==
<div data-type="note" data-timestamp="0:00:01">
Eric Weinstein interviews Sir Roger Penrose, Episode 20 of The Portal
</div>
<div data-type="note" data-timestamp="0:00:01">
Player by Demp
Annotations by R1chard5mith
</div>
<div data-type="resource" data-timestamp="0:00:01">
[[File:Eric weinstein.png|thumb]]
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<div data-type="resource" data-timestamp="0:00:02">
[[File:Roger penrose.jpg|thumb]]
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<div data-type="resource" data-timestamp="0:11:05">
https://upload.wikimedia.org/wikipedia/en/thumb/f/f9/Sciama2.jpg/200px-Sciama2.jpg
[https://en.wikipedia.org/wiki/Dennis_W._Sciama Dennis Sciama]
</div>
<div data-type="resource" data-timestamp="0:11:10">
https://static.scientificamerican.com/blogs/cache/file/7DAF801B-9AED-4B63-B7EAD43F5E9B1B2D_source.jpg
</div>
<div data-type="note" data-timestamp="0:11:17">
[https://en.wikipedia.org/wiki/Physics Physics]
</div>
<div data-type="resource" data-timestamp="0:11:31">
https://upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Stephen_Hawking.StarChild.jpg/220px-Stephen_Hawking.StarChild.jpg
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<div data-type="note" data-timestamp="0:11:31">
[https://en.wikipedia.org/wiki/Stephen_Hawking Stephen Hawking]
</div>
<div data-type="resource" data-timestamp="0:11:40">
https://upload.wikimedia.org/wikipedia/commons/thumb/5/50/Paul_Dirac%2C_1933.jpg/220px-Paul_Dirac%2C_1933.jpg
</div>
<div data-type="note" data-timestamp="0:11:40">
[https://en.wikipedia.org/wiki/Paul_Dirac Paul Dirac]
</div>
<div data-type="note" data-timestamp="0:11:50">
Book: [https://www.amazon.com/Strangest-Man-Hidden-Dirac-Mystic/dp/0465022103 The Strangest Man: The Hidden Life of Paul Dirac, Mystic of the Atom] by [https://en.wikipedia.org/wiki/Graham_Farmelo Graham Farmelo]
</div>
<div data-type="note" data-timestamp="0:12:22">
[https://en.wikipedia.org/wiki/Albert_Einstein Albert Einstein]
<div>
<div data-type="note" data-timestamp="0:12:35">
Eric imputes that Diracā€™s hair was not as good as Einsteinā€™s.
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<div data-type="resource" data-timestamp="0:12:34">
https://vignette.wikia.nocookie.net/wildcards/images/c/c7/Einstein.jpg
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<div data-type="note" data-timestamp="0:12:42">
[https://en.wikipedia.org/wiki/Quantum_mechanics Quantum mechanics]
</div>
<div data-type="resource" data-timestamp="0:12:42">
https://upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Hydrogen_Density_Plots.png/290px-Hydrogen_Density_Plots.png
</div>
<div data-type="resource" data-timestamp="0:13:00">
http://arkadiusz-jadczyk.eu/blog/wp-content/uploads/2017/01/Penrose-Rindler-Clifford-parallels.jpg
</div>
<div data-type="note" data-timestamp="0:13:08">
NB:Roger Penrose and Eric Weinstein are friends with the guy who does the commentary for the cage fighting. What a time to be alive.
[https://www.youtube.com/channel/UCzQUP1qoWDoEbmsQxvdjxgQ Joe Rogan Experience]
</div>
<div data-type="resource" data-timestamp="0:13:08">
https://i.kym-cdn.com/entries/icons/original/000/027/944/everdonedmt.jpg
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<div data-type="resource" data-timestamp="0:13:11">
https://thumbs.gfycat.com/GrotesqueApprehensiveCusimanse-mobile.mp4
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<div data-type="note" data-timestamp="0:13:11">
The [https://en.wikipedia.org/wiki/Hopf_fibration Hopf Fibration] is the only non-trivial principal bundle that can be visually seen
</div>
<div data-type="resource" data-timestamp="0:13:11">
https://upload.wikimedia.org/wikipedia/commons/thumb/b/b9/Hopf_Fibration.png/250px-Hopf_Fibration.png
</div>
<div data-type="note" data-timestamp="0:13:13">
[https://en.wikipedia.org/wiki/Fiber_bundle Fibre Bundles]
</div>
<div data-type="note" data-timestamp="0:13:13">
The ā€œ[https://en.wikipedia.org/wiki/Hopf_fibration Hopf fibration]ā€, or the ā€œ[https://en.wikipedia.org/wiki/Clifford_parallel Clifford Parallels]ā€ was instrumental in the subject of Twistor Theory.
</div>
<div data-type="note" data-timestamp="0:13:20">
Thereā€™s also this diagram, which I found at a blog here: http://arkadiusz-jadczyk.eu/blog/tag/penrose/
</div>
<div data-type="resource" data-timestamp="0:13:20">
http://arkadiusz-jadczyk.eu/blog/wp-content/uploads/2017/01/Penrose-Rindler-Clifford-parallels.jpg
</div>
<div data-type="note" data-timestamp="0:13:22">
And thereā€™s this diagram, which I found at this link ( http://users.ox.ac.uk/~tweb/00001/ ) which is an HTML presentation of ā€œOn the Origins of Twistor Theoryā€ - Roger Penrose, 1987
</div>
<div data-type="resource" data-timestamp="0:13:22">
http://users.ox.ac.uk/~tweb/00001/robinson.jpg
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<div data-type="resource" data-timestamp="0:14:17">
[[File:Road to reality hopf.jpg|thumb]]
</div>
<div data-type="note" data-timestamp="0:14:39">
[https://en.wikipedia.org/wiki/Spinor Spinors] (Wikipedia)
[https://arxiv.org/pdf/1312.3824.pdf An introduction to spinors by Andrew M Steane] (pdf)
[https://www.youtube.com/playlist?list=PLJHszsWbB6hoOo_wMb0b6T44KM_ABZtBs Spinors for Beginners by Eigenchris] (YouTube course)
</div>
<div data-type="resource" data-timestamp="0:14:39">
[[File:Spinors flag.png|thumb]]
</div>
<div data-type="note" data-timestamp="0:14:43">
[https://en.wikipedia.org/wiki/Twistor_theory Twistor Theory]
</div>
<div data-type="note" data-timestamp="0:14:54">
quantum mechanics and the first course
</div>
<div data-type="note" data-timestamp="0:14:59">
Dirac gave a course (2 courses) of lectures on Quantum Mechanics
Course 1 - Basic [https://en.wikipedia.org/wiki/Quantum_mechanics Quantum Mechanics]
Course 2 - [https://en.wikipedia.org/wiki/Quantum_field_theory Quantum Field Theory] but also Spinors
</div>
<div data-type="resource" data-timestamp="0:15:00">
https://upload.wikimedia.org/wikipedia/commons/thumb/c/c1/Spinor_on_the_circle.png/330px-Spinor_on_the_circle.png
</div>
<div data-type="note" data-timestamp="0:15:02">
In the second course (when Penrose took the course) Dirac deviated from his normal course of lectures to give two or three lectures on [https://en.wikipedia.org/wiki/Spinor#Component_spinors Two Component Spinors]. ([https://arxiv.org/abs/1312.3824 Spinors])
</div>
<div data-type="note" data-timestamp="0:16:06">
[https://en.wikipedia.org/wiki/Quark Quarks]
</div>
<div data-type="resource" data-timestamp="0:16:06">
https://upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Proton_quark_structure.svg/225px-Proton_quark_structure.svg.png
</div>
<div data-type="note" data-timestamp="0:16:12">
General phenomenon of [https://en.wikipedia.org/wiki/Wave Waves]
</div>
<div data-type="resource" data-timestamp="0:16:12">
https://upload.wikimedia.org/wikipedia/commons/thumb/c/c8/Superpositionprinciple.gif/220px-Superpositionprinciple.gif
</div>
<div data-type="resource" data-timestamp="0:16:39">
[[File:Cordon spinors book.png|thumb]]
</div>
<div data-type="note" data-timestamp="0:16:39">
Denis recommended to Roger a book by Corson (presumably this one: [https://www.amazon.com/Introduction-Tensors-Spinors-Relativistic-Wave-Equations/dp/B0000CIMO7 Introduction to Tensors, Spinors, and Relativistic Wave-Equations], 1953 by [https://www.ias.edu/scholars/edward-michael-corson E.M. Corson] *
</div>
<div data-type="note" data-timestamp="0:17:39">
[https://en.wikipedia.org/wiki/Quantum_field_theory Quantum Field Theory]
</div>
<div data-type="resource" data-timestamp="0:17:39">
https://upload.wikimedia.org/wikipedia/commons/thumb/1/1f/Feynmann_Diagram_Gluon_Radiation.svg/211px-Feynmann_Diagram_Gluon_Radiation.svg.png
</div>
<div data-type="resource" data-timestamp="0:18:00">
https://www.chu.cam.ac.uk/media/assets/8c/c2317c444c70a04633e4fd29095ef1adda7d8f.jpg
</div>
<div data-type="note" data-timestamp="0:18:26">
Other people said that Diracā€™s course was just like his book but Roger hadnā€™t read the book.
(I assume this one? Dirac - [https://www.amazon.com/Principles-Quantum-Mechanics-P-Dirac/dp/1607965607 Principles of Quantum Mechanics])
</div>
<div data-type="note" data-timestamp="0:18:55">
Mention of Mathematicians:
* [https://en.wikipedia.org/wiki/Wilhelm_Killing Wilhelm Killing]
* ā€œLeeā€? - this Lee https://en.wikipedia.org/wiki/John_M._Lee was in Differential Geometry but was born in 1950. Maybe his father?
* [https://en.wikipedia.org/wiki/%C3%89lie_Cartan Ɖlie Cartan]
</div>
<div data-type="resource" data-timestamp="0:18:55">
https://upload.wikimedia.org/wikipedia/commons/thumb/5/57/Wilhelm_Karl_Joseph_Killing.jpeg/220px-Wilhelm_Karl_Joseph_Killing.jpeg
Wilhelm Killing
</div>
<div data-type="resource" data-timestamp="0:18:55">
https://upload.wikimedia.org/wikipedia/en/thumb/e/e8/Elie_Cartan.jpg/220px-Elie_Cartan.jpg
Elie Cartan
</div>
<div data-type="resource" data-timestamp="0:19:00">
[[File:Eric gesticulating.png|thumb]]
</div>
<div data-type="note" data-timestamp="0:19:17">
Sir Roger Penroseā€™s favourite film is [https://en.wikipedia.org/wiki/2001:_A_Space_Odyssey_(film) 2001: A Space Odyssey]
</div>
<div data-type="resource" data-timestamp="0:19:15">
https://upload.wikimedia.org/wikipedia/en/thumb/1/11/2001_A_Space_Odyssey_%281968%29.png/220px-2001_A_Space_Odyssey_%281968%29.png
</div>
<div data-type="resource" data-timestamp="0:19:20">
[[File:2001 monolith.png|thumb]]
</div>
<div data-type="resource" data-timestamp="0:19:25">
https://vignette.wikia.nocookie.net/optical-illusions/images/6/62/Note.gif
</div>
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<!--
[NOTE: Iā€™m not sure if he ever gets back to saying what this proposal is. It looks like it might be ā€œConformal Cyclic Cosmologyā€, see eg; [https://physicsworld.com/a/new-evidence-for-cyclic-universe-claimed-by-roger-penrose-and-colleagues/ Physics World], [https://physicsworld.com/a/inside-penroses-universe/ ibid], his own book, [https://en.wikipedia.org/wiki/Cycles_of_Time Cycles of Time]]
8:00 = Talking to his brother in the Kingswood Restaurant, Cambridge *, Roger expressed his doubts and was referred to Dennis.
[NOTE: I tried to find a link for this restaurant, which appears to no longer exist, and came across this really interesting paper by Professor Penrose and [https://en.wikipedia.org/wiki/George_F._R._Ellis George Ellis], which is a kind of ā€œscientific eulogyā€* for Dennis Sciama, in which the same anecdote is recalled, amongst others: https://royalsocietypublishing.org/doi/pdf/10.1098/rsbm.2009.0023 (pdf) ]
&lt;&lt;&lt;Thereā€™s probably a better term for this, right?
&lt;&lt;&lt;When I search for ā€˜ā€œthe kingswood restaurantā€ cambridgeā€™ I donā€™t turn up anything that seems relevant and when I add the word ā€œrememberā€ to that search I start to turn up links to Sir Roger himself.Possibly it was called by a different name. Also possible that no trace of it has made it onto the internet other than his telling.
Penroseā€™s supervisor was Hodge - [https://en.wikipedia.org/wiki/W._V._D._Hodge W.V.D Hodge]
But later he threw Roger out and Todd became his supervisor - [https://en.wikipedia.org/wiki/J._A._Todd J.A Todd] *
&lt;&lt;&lt;The links Iā€™ve added to Hodge and Todd both seem right but Iā€™m not personally familiar
The Hopf Fibration ([https://en.wikipedia.org/wiki/Hopf_fibration#/media/File:Hopf_Fibration.png source],[https://nilesjohnson.net/hopf.html gif])
PEN - The ā€œ[https://en.wikipedia.org/wiki/Hopf_fibration Hopf fibration]ā€, or the ā€œ[https://en.wikipedia.org/wiki/Clifford_parallel Clifford Parallels]ā€ was instrumental in the subject of Twistor Theory.
14:00 = Penroseā€™s diagram
There were three versions. The third version is in The Road to Reality. He thinks the second version is probably the best.
(I think they are talking about the diagram of the Hopf Fibration ?? as seen at the link above )
I '''think''' this is the one from ā€œThe Road to Realityā€, which would make it Version 3: * **
please correct this if you know better
Thereā€™s also this diagram, which I found at a blog here: http://arkadiusz-jadczyk.eu/blog/tag/penrose/
And thereā€™s this diagram, which I found at this link ( http://users.ox.ac.uk/~tweb/00001/ ) which is an HTML presentation of ā€œOn the Origins of Twistor Theoryā€ - Roger Penrose, 1987
[NOTE: these latter two might be Versions 1 and 2? Or later reproductions.]
Penrose thinks Version 2 was the best.
In the second course (when Penrose took the course) Dirac deviated from his normal course of lectures to give two or three lectures on [https://en.wikipedia.org/wiki/Spinor#Component_spinors Two Component Spinors]. ([https://arxiv.org/abs/1312.3824 Spinors])
Denis recommended to Roger a book by Corson (presumably this one: [https://www.amazon.com/Introduction-Tensors-Spinors-Relativistic-Wave-Equations/dp/B0000CIMO7 Introduction to Tensors, Spinors, and Relativistic Wave-Equations], 1953 by [https://www.ias.edu/scholars/edward-michael-corson E.M. Corson] *
The book was reviewed by [https://en.wikipedia.org/wiki/Abdus_Salam Abdus Salam] in 1955 (https://www.nature.com/articles/175831b0 )
(I assume this one? Dirac - [https://www.amazon.com/Principles-Quantum-Mechanics-P-Dirac/dp/1607965607 Principles of Quantum Mechanics])
Taken from Introduction to [https://arxiv.org/pdf/1312.3824.pdf Spinors - Andrew M Steane 2013] (pdf)
WEIN: [https://en.wikipedia.org/wiki/Klein_bottle Klein Bottle]:
== Interlude: Klein Bottles ==
Get your Klein Bottle today from our friends at [https://www.kleinbottle.com/ Acme Klein Bottles]
Hereā€™s a good [https://www.youtube.com/watch?v=AAsICMPwGPY video about Klein Bottles]
(the presenter is [https://en.wikipedia.org/wiki/Clifford_Stoll Clifford Stoll], Astronomer and proprietor of [https://www.kleinbottle.com/ Acme Klein Bottles] )
* The Klein Bottle is ā€œtwo [https://en.wikipedia.org/wiki/M%C3%B6bius_strip Mƶbius Strips] stitched togetherā€ (after [https://en.wikipedia.org/wiki/August_Ferdinand_M%C3%B6bius August Ferdinand Mƶbius])
* The Klein Bottle is named after [https://en.wikipedia.org/wiki/Felix_Klein Felix Klein]
* The Klein Bottle only has one side
* Klein bottles is 3D Universes must have a self-intersection
* Klein hats are continuously deformable back to themselves
* Cliffordā€™s friend, [https://en.wikipedia.org/wiki/Robert_J._Lang Robert Lang], made an [https://langorigami.com/crease-pattern/klein-bottle-opus-444/ Origami Klein Bottle] *
Here is another, related video, also from Numberphile, about the [https://www.youtube.com/watch?v=3_VydFQmtZ8&list=PLt5AfwLFPxWIpgtcFs_7fHGUedGEKu73p&index=8&t=0s Topology of a Twisted Torus]
&lt;&lt;&lt;NB: This is a link to the actual pattern so that you can make your very own Origami Klein bottle.
This video is about slicing up toroids. The presenter is [https://en.wikipedia.org/wiki/Carlo_H._S%C3%A9quin Carlo H. SĆ©quin]. Later in the podcast, Roger talks about the [https://en.wikipedia.org/wiki/Clifford_parallel Clifford Parallels] dividing up space in a similar way. This helped me to visualise that.
&lt;&lt;&lt;One of his sculptures is not far from where I live. May have to make a visit.
It also introduced me to [http://www2.memenet.or.jp/~keizo/index.html Keizo Ushio] who makes amazing toroidal sculptures, like this one *
([http://www2.memenet.or.jp/~keizo/NiihamaSculptureProject.htm source])
Here is an interview with [https://www.youtube.com/watch?v=bkInMmWcblI Keizo Ushio].
In [https://www.youtube.com/watch?v=_w1qkiRHQ4E this video] he can be heard speaking in English.
There are some more interesting (if somewhat amateur) visualisations in [https://www.youtube.com/watch?v=sRTKSzAOBr4 this Youtube video], from which I learned:
* The Klein Bottle is a [http://mathworld.wolfram.com/NonorientableSurface.html non-orientable], [https://www2.cs.duke.edu/courses/fall06/cps296.1/Lectures/sec-II-1.pdf 2-dimensional manifold].
* It can be ā€œ[https://en.wikipedia.org/wiki/Immersion_(mathematics) immersed]ā€ into [https://en.wikipedia.org/wiki/Three-dimensional_space Euclidean 3-dimensional space] with a [https://en.wikipedia.org/wiki/Intersection_theory self-intersection].
* Non-orientable means ā€œthere exists no continuous normal unit vector fieldā€
[Note: googling for ā€œthe square root of the Klein Bottleā€ didnā€™t get me far but searching for ā€œ[https://en.wikipedia.org/wiki/Double_cover double cover]ā€ I got useful things like [https://math.stackexchange.com/questions/1073425/two-sheeted-covering-of-the-klein-bottle-by-the-torus this question on math Stackexchange], where someone has drawn this:
And the top respondent says ā€œMost topologists would be happy just drawing the diagram youā€™ve drawnā€ (to prove that there is a two-sheeted covering of the Klein bottle by the Torus)
Reading that answer and then this one: https://math.stackexchange.com/questions/140439/torus-as-double-cover-of-the-klein-bottle gave me a pretty good idea of whatā€™s going on.]
Dirac gave a demonstration due to [https://en.wikipedia.org/wiki/Hermann_Weyl Hermann Weyl] of rolling one [https://en.wikipedia.org/wiki/Cone cone] on another
Here is a google-books link to Penrose describing the same model in the [https://books.google.com.au/books?id=Qk5Q74166qcC&pg=PA41&lpg=PA41#v=onepage&q&f=false Hermann Weyl Centenary Lectures].
[http://www.weylmann.com This site] also has a description of the model (http://www.weylmann.com/2010archive.shtml - you need to search for the word ā€œconeā€ to find the right article) and lots of other information about Weyl himself. It includes this diagram to illustrate the model:
The author of the site is William O. Straub and he has written other papers about Spinors, including eg; [http://www.weylmann.com/weyldirac.pdf Weyl Spinors and Diracā€™s Electron Equation].
This [https://www.youtube.com/watch?v=yLH4l_SoIy0 video on Youtube] is a visualisation of rolling one coin around another.
Diracā€™s scissors, aka the [https://en.wikipedia.org/wiki/Plate_trick Plate Trick], (related to?) the [https://www.youtube.com/watch?v=JaIR-cWk_-o&feature=youtu.be Belt Trick]
Air on a Dirac String: https://www.youtube.com/watch?v=CYBqIRM8GiY
34:00 = [https://en.wikipedia.org/wiki/Phlogiston_theory Phlogiston]
[NOTE: Phlogiston was the supposed substance that inhered in bodies capable of combustio
Book: [https://en.wikipedia.org/wiki/Charles_Darwin Darwin] - ON THE VARIOUS CONTRIVANCES BY WHICH BRITISH AND FOREIGN ORCHIDS ARE FERTILISED BY INSECTS, AND ON THE GOOD EFFECTS OF INTERCROSSING. BY CHARLES DARWIN, M.A., F.R.S., &amp;c.Ā WITH ILLUSTRATIONS.
([http://darwin-online.org.uk/content/frameset?itemID=F800&viewtype=text&pageseq=1 full text as HTML])
38:00 = [https://en.wikipedia.org/wiki/Ascending_and_Descending MC ESCHER - Ascending and Descending] (The Penrose Stairs)
Ascending and Descending - M.C. Escher
39:00 = PEN - anecdote about visiting Amsterdam. Mentions [https://en.wikipedia.org/wiki/Shaun_Wylie Shaun Wyile] *
41:00 = ā€œI played around with this and whittled it down to the triangle which people refer to as a ā€˜tribarā€™.ā€ NB: Heā€™s being modest and we actually call it a [https://en.wikipedia.org/wiki/Penrose_triangle Penrose Triangle]. ([http://mathworld.wolfram.com/PenroseTriangle.html Penrose Triangle on Wolfram])
Iā€™m pretty sure this is the paper: [https://onlinelibrary.wiley.com/doi/abs/10.1111/j.2044-8295.1958.tb00634.x impossible objects a special type of visual illusion] - L.S. Penrose and R. Penrose *
&lt;&lt;&lt;NB: Wiley want $7 to rent this 62 year old paper for 48 hours (!) or $42 (!!) to buy it as a PDF. It is on SciHub and is 3 pages long. If you had to buy Penroseā€™s (paperback) book (tRtR) for the same per-page price, it would cost $14,000 a copy
Escher gave Penrose a print and it is in the [https://en.wikipedia.org/wiki/Ashmolean_Museum Ashmolean Museum]
&lt;&lt;&lt;I canā€™t read this article: https://www.thetimes.co.uk/article/roger-penrose-on-his-friend-mc-escher-the-genius-that-galleries-ignored-90nhp8gsd0l because itā€™s behind a paywall, but the google link-summary says ā€œā€¦ and he chose Fish and Scales, now on loan to the Ashmolean in Oxfordā€
(I think) the print was [https://www.wikiart.org/en/m-c-escher/fishes-and-scales Fishes and Scales] *
[https://en.wikipedia.org/wiki/Roman_Jackiw Roman Jackiw] - ā€œwhen we talked to the geometers, we started to learn new thingsā€
Jakiw is interviewed [https://www.aip.org/history-programs/niels-bohr-library/oral-histories/34449 here at aip] *
[https://people.maths.ox.ac.uk/lmason/Tn/ Twistor Newsletter]
* [https://en.wikipedia.org/wiki/Sheldon_Lee_Glashow Sheldon Lee Glashow]
* [https://en.wikipedia.org/wiki/Howard_Georgi Howard Georgi]
* [https://en.wikipedia.org/wiki/Jean-Pierre_Petit Jean-Pierre Petit] *
* [https://en.wikipedia.org/wiki/Abdus_Salam Abdus Salam]
* [https://en.wikipedia.org/wiki/Engelbert_Sch%C3%BCcking Engelbert SchĆ¼cking]
* [https://en.wikipedia.org/wiki/Roy_Kerr Roy Kerr]
* [https://en.wikipedia.org/wiki/Rainer_K._Sachs Rainer K. Sachs]
He travelled back with [https://www.researchgate.net/scientific-contributions/2038878487_Istvan_Ozsvath IstvĆ”n OzsvĆ”th] * who didnā€™t speak much and started to think about
I visited (Bruno) [https://en.wikipedia.org/wiki/Bruno_Zumino Zumino]
Deep supersymmetric model (the [[https://en.wikipedia.org/wiki/Wessā€“Zumino_model|Wess-Zumino model]])
* [https://en.wikipedia.org/wiki/Nigel_Hitchin Nigel Hitchin]
* Mason - possibly [https://www.maths.ox.ac.uk/people/lionel.mason Lionel Mason]?
* [https://www.maths.ox.ac.uk/people/philip.candelas Philip Candelas]
-->
= Algebraic Geometry of Computing =
= Algebraic Geometry of Computing =
Finite state machines appear in a variety of instantiations: mechanical, electronic, fluidic. The physical mechanisms involved necessitate that the design is described by differential equations, but ultimately the manipulation of abstracted "logical" states is the final goal. Thus we can describe the architecture of a general finite state machine with <math> \mathbb{Z}/2\mathbb{Z} </math> algebra (or other finite rings too).
Finite state machines appear in a variety of instantiations: mechanical, electronic, fluidic. The physical mechanisms involved necessitate that the design is described by differential equations, but ultimately the manipulation of abstracted "logical" states is the final goal. Thus we can describe the architecture of a general finite state machine with <math> \mathbb{Z}/2\mathbb{Z} </math> algebra (or other finite rings too).
Retrieved from "https://theportal.wiki/wiki/Special:MobileDiff/13763"