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Due to my poor mathematical knowledge, I have difficulty being "formally precise", but here is an informal attempt at explaining what I mean by "form":

Form:

  1. How can I represent any imaginable shape (1 dimension, 2 dimensional, or n...dimensional) mathematically?
  2. How can I model the physics behind material and shape?
  3. What is the mathematical connection between material, shape and symmetry?
  4. How can I model "shape transformation" (e.g. origami, folding, etc.)?
  5. What is the mathematics behind packings, tilings, coverings etc.?

In general, what do I need to learn in order to be able to approach problems that this lab deals with?

My current mathematical knowledge:

  1. University level calculus.
  2. ODEs
  3. linear algebra
bzm3r
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  • Topology or Differential Geometry, take your pick – Lemon Jan 25 '14 at 05:55
  • @sidht Could you explain why both are relevant? – bzm3r Jan 25 '14 at 06:04
  • Your question is a little silly: there is no well-defined mapping from the subjects you ask about to mathematical disciplines. All the subjects you mention draw on a huge number of different mathematical fields and concepts: abstract algebra especially (group theory is the language for symmetry considerations), topology (which deals abstractly with what 'shapes' are possible, after carefully interpreting the word 'shape' in different, precise ways), even combinatorics (for packings and tilings). The simplest answer about what you need to know for that Harvard lab is: get a math degree. – symplectomorphic Jan 25 '14 at 06:14
  • If there were a simple answer to "what math do I need to know to do understand the physics 'behind material and shape'?" all the mathematical and theoretical physicists in the world would be out of jobs. – symplectomorphic Jan 25 '14 at 06:16
  • @symplectomorphic I didn't ask the question: "what math do I need to know to do understand the physics 'behind material and shape'?". I asked the question: "what do I need to learn in order to be able to approach problems that deal with material and shape?". I would like to be able to participate in the process of discovering the mathematics behind these phenomena; I am just not quite sure where to start. – bzm3r Jan 25 '14 at 06:21
  • you asked "How can I model the physics behind material and shape?" the answer is: there is no simple answer (it's not even clear what "the physics behind material and shape" means). if you're simply asking what kind of subjects you need to start studying in order to eventually be able to think about physics, symmetry, folding, and packing (which I'm trying to tell you is a hugely diverse set of topics), you need to take: abstract algebra, topology, geometry, and a whole lot more. in other words: you need to study undergraduate mathematics. – symplectomorphic Jan 25 '14 at 06:27
  • @twirlobite: I see while you improved your question from the version originally posted on MathOverflow, it is still in danger of being closed. I strongly suspect that the reason is that it is still too broad and difficult to answer well. Can you narrow it down by any chance? – J W Jan 25 '14 at 17:30
  • @JW I am considering just deleting it myself. I am not sure how to narrow it down further. This question was actually the original version, and I removed information about my current mathematical background on MathOverflow in order to make less about me, but it seems to not have been an improvement. – bzm3r Jan 25 '14 at 17:35
  • @twirlobite: Perhaps you could pick a very specific topic that the lab tackles, or choose a fairly precise area of mathematics? Furthermore, provide clear motivation and, if possible, show what you have already done to answer the question. It might help. – J W Jan 25 '14 at 17:43
  • @JW Thank you very much for that constructive criticism. I think as it stands right now, I will need to work more on seeing how I can answer the question myself, before I make another post. For that reason, I'll delete the MathOverflow post. – bzm3r Jan 25 '14 at 17:50
  • @twirlobite: May I wish you all the very best. – J W Jan 25 '14 at 17:58
  • @JW Thank you. By the way, I'd just like to say that your suggestion of Devadoss and O'Rourke's book has been fantastic. I ended up finding O'Rourke's MO and M.SE pages. His posts were an absolute goldmine of information for the kind of stuff I am interested in! – bzm3r Jan 25 '14 at 18:23
  • @twirlobite Your question is probably appropriate for the nearly-in-beta-SE http://area51.stackexchange.com/proposals/64216/mathematics-learning-studying-and-education. Check out the proposal and commit to it if you're interested. Then we can get it off the ground and get the site in beta! – Xoque55 Mar 03 '14 at 04:44
  • @Xoque55 I am signed up for that proposal very early on, and also proposed it as a question in the define stage: http://area51.stackexchange.com/proposals/64216/mathematics-learning-studying-and-education – bzm3r Mar 03 '14 at 05:43

1 Answers1

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It could be helpful to delve into discrete and computational geometry. One way to get started is with Devadoss & O'Rourke's Discrete and Computational Geometry.

For folding in particular, try O'Rourke's How to Fold It for a gentle introduction and Demaine & O'Rourke's Geometric Folding Algorithms for more advanced coverage. As you progress, you may find yourself needing to learn some combinatorics, graph theory, algorithms and/or probability.

However, to cover all the things you have mentioned, you'll need a very wide range of mathematics, as stated in the comments on your question.

J W
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  • You could also find the following links useful: https://en.wikipedia.org/wiki/Shape_analysis Specifically, a new theory within statistics is statistical shape analysis: https://en.wikipedia.org/wiki/Statistical_shape_analysis The foillowing link will be of interest to you: http://www.dam.brown.edu/people/mumford/vision/shape.php – kjetil b halvorsen Jan 25 '14 at 13:57
  • @kjetilbhalvorsen Thank you. Also, to add another resource for the sake of posterity: The shape of nature and the nature of shape, by L. Mahadevan – bzm3r Jan 25 '14 at 19:04
  • @twirlobite: You may also be interested in Forced Crumpling. – J W Jan 26 '14 at 16:53
  • @JW I was looking at some of your comments (along with J. O'Rourke's stuff) on MO, and I noticed you made the following comment: http://mathoverflow.net/a/83890/45695

    Would your recommendation for the Visual Group Theory book help fill in the gaps there?

     – bzm3r Jan 29 '14 at 01:14
  • @JW Or would a combination of Crossley's book along with O'Rourke's suggested book suit me best? – bzm3r Jan 29 '14 at 01:16
  • @twirlobite: If you'd like to learn topology with a taster of its numerous applications then Introduction to Topology: Pure and Applied is a good choice. It would probably be good alongside or after Devadoss & O'Rourke. However, if you want to move into algebraic or computational topology then at some point you'll need to pick up group theory, which is worth learning for other reasons as well. Visual Group Theory is a nice intuitive way to get started. – J W Jan 29 '14 at 08:01
  • @JW I am loving those books so far! I also found another neat little resource: The Mathematics of Soap Films, by Oprea – bzm3r Jan 30 '14 at 22:21
  • @JW I don't know if that book is too good in terms of quality (the explanations for some things aren't too clear to me), but it does provide an interesting roadmap. – bzm3r Jan 30 '14 at 23:05
  • @twirlobite: Thanks for the suggested book on soap films. By the way, I'm not sure which book you're referring to when you write "I don't know if that book is too good in terms of quality." – J W Jan 31 '14 at 06:51
  • @JW I was referring to the soap film book...I was a little unhappy with the derivation of the Young-Laplace equation. I don't know, it didn't seem clear to me. – bzm3r Jan 31 '14 at 07:28