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These methods were invented decades ago by Penrose (1971) [2], as a way of visualizing and improving on Einstein's summation convention [1]. Similar "string dia

These methods were invented decades ago by Penrose (1971) [2], as a way of visualizing and improving on Einstein's summation convention [1]. Similar "string diagrams" were popularized for various categories by John Baez in "This Week's Finds" (TWF) [3], with extensive applications to QM by Coecke [4]. There are other modern examples all over the interweb [5] ... [1] https://en.wikipedia.org/wiki/Einstein_notation https://en.wikipedia.org/wiki/Einstein_notation [2] https://en.wikipedia.org/wiki/Penrose_graphical_notation https://en.wikipedia.org/wiki/Penrose_graphical_notation [3] http://math.ucr.edu/home/baez/week79.html http://math.ucr.edu/home/baez/week79.html ff [4] https://www.cs.ox.ac.uk/ss2014/programme/Bob.pdf https://www.cs.ox.ac.uk/ss2014/programme/Bob.pdf [5] https://graphicallinearalgebra.net/ https://graphicallinearalgebra.net/

fantispug 7y ago
They do mention Penrose notation in the linked arxiv paper[1], as well as Cvitanović's Birdtracks[2] which let you do calculations on representations of groups. They claim they're making it more accessible with pedagogical examples (and innovations in representing differential vector calculus). I still prefer index notation. Their notation looks pretty intricate and I think it would be really hard to do long calculations in it. It's also not clear to me how you'd generalise it to non-commutative vector algebras like in quantum mechanics. [1] https://arxiv.org/pdf/1911.00892.pdf https://arxiv.org/pdf/1911.00892.pdf [2] http://birdtracks.eu/ http://birdtracks.eu/
kmill 7y ago
The conclusions section has a nice short history of some graphical notations, noting "graphical notations of tensor algebra have a history spanning over a century." Equation 18 in their paper is in Penrose's original paper ("Applications of negative dimensional tensors"). Fun fact: if you take a planar graph, all of whose vertices are degree-3, then interpret them as triple-products/determinants/cross-products, then the absolute value of the resulting scalar is proportional to the number of proper 4-colorings of the regions complementary to the graph. A reiteration of this is in Bar-Natan's paper "Lie algebras and the four color theorem." However, their contribution seems to be notation for dealing with 3-dimensional derivatives (gradient, curl, and divergence), which are special due to Hodge duality and the existence of a triple product. Equation 30 implying curl of gradient and divergence of curl both being zero is pretty nice. I think the authors are correct that the specialization of the notation for 3D vector calculus had not been written up yet.
- mikhailfranco 7y ago
  Yes, your fun fact is in Baez's TWF week 92 (1996): http://math.ucr.edu/home/baez/week92.html http://math.ucr.edu/home/baez/week92.html and the core proposition goes back to Penrose.
  - kmill 7y ago
    I see I was rather oblique, but I brought up the fun fact because that equation is how Penrose proved it (to account for signs). By the way, the TWF article is citing the Bar-Natan paper.
eternalban 7y ago
Penrose's paper: https://mathstat.dal.ca/~selinger/papers/papers/graphical-bib/public/Penrose-applications-of-negative-dimensional-tensors.pdf https://mathstat.dal.ca/~selinger/papers/papers/graphical-bi...

fantispug 7y ago

They do mention Penrose notation in the linked arxiv paper[1], as well as Cvitanović's Birdtracks[2] which let you do calculations on representations of groups. They claim they're making it more accessible with pedagogical examples (and innovations in representing differential vector calculus). I still prefer index notation. Their notation looks pretty intricate and I think it would be really hard to do long calculations in it. It's also not clear to me how you'd generalise it to non-commutative vector algebras like in quantum mechanics. [1] https://arxiv.org/pdf/1911.00892.pdf https://arxiv.org/pdf/1911.00892.pdf [2] http://birdtracks.eu/ http://birdtracks.eu/

kmill 7y ago

The conclusions section has a nice short history of some graphical notations, noting "graphical notations of tensor algebra have a history spanning over a century." Equation 18 in their paper is in Penrose's original paper ("Applications of negative dimensional tensors"). Fun fact: if you take a planar graph, all of whose vertices are degree-3, then interpret them as triple-products/determinants/cross-products, then the absolute value of the resulting scalar is proportional to the number of proper 4-colorings of the regions complementary to the graph. A reiteration of this is in Bar-Natan's paper "Lie algebras and the four color theorem." However, their contribution seems to be notation for dealing with 3-dimensional derivatives (gradient, curl, and divergence), which are special due to Hodge duality and the existence of a triple product. Equation 30 implying curl of gradient and divergence of curl both being zero is pretty nice. I think the authors are correct that the specialization of the notation for 3D vector calculus had not been written up yet.

mikhailfranco 7y ago
Yes, your fun fact is in Baez's TWF week 92 (1996): http://math.ucr.edu/home/baez/week92.html http://math.ucr.edu/home/baez/week92.html and the core proposition goes back to Penrose.
- kmill 7y ago
  I see I was rather oblique, but I brought up the fun fact because that equation is how Penrose proved it (to account for signs). By the way, the TWF article is citing the Bar-Natan paper.

eternalban 7y ago

Penrose's paper: https://mathstat.dal.ca/~selinger/papers/papers/graphical-bib/public/Penrose-applications-of-negative-dimensional-tensors.pdf https://mathstat.dal.ca/~selinger/papers/papers/graphical-bi...