Wikipedia:Reference desk/Archives/Mathematics/2017 January 12

Welcome to the Wikipedia Mathematics Reference Desk Archives
The page you are currently viewing is an archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages.

January 12

Traditional Chinese mathematics?

At the top of our article on Principia Mathematica is a quote "He [Russell] said once, after some contact with the Chinese language, that he was horrified to find that the language of Principia Mathematica was an Indo-European one". This got me thinking - China and Greece developed quite advanced mathematics more or less independently, but did Chinese mathematics differ at all in its axioms and principles from Western mathematics? From what I can tell of the article Chinese mathematics, they got the same results in geometry and algebra, for instance, and they independently derived Pascal's triangle. Are there any problems that can be solved by the Chinese framework but not the western one? Smurrayinchester 10:12, 12 January 2017 (UTC)

Your question is framed in a way that the Chinese would have understood but misses out what Russell was on about. Their mathematics was in the main concerned about the how and what rather than the why. The Greeks concentrated on the why - sometimes to the extent of overlooking useful ways of doing things which didn't have a good formal basis like for instance The Method of Mechanical Theorems of Archimedes, though even that mentioned theorems which weren't a big thing with the Chinese. Dmcq (talk) 14:05, 12 January 2017 (UTC)

It's not clear in what sense PM was meant to be an Indo-European language; it could be something like grammatical word order rather than the axiomatic method. It's doubtful to me whether axioms in the sense we use them now were even used in traditional Chinese mathematics. Which is not to say they were used consistently in Europe either. As far as I know there were no formal axioms for number theory until Peano, and even in Euclid, the historic model for the axiomatic method, there is famously a logical error right off the bat in Book 1 Proposition 1. --RDBury (talk) 17:54, 12 January 2017 (UTC)

He didn't mean it to be. He was horrified to find the way of thought was foreign to the Chinese. We grow up with the language of our culture and if a concept is awkward in the language we tend not to use it. Why is a straightforward word in English just like how or what. In Chinese saying why is longer than things like who or what or how or where. It takes three Chinese characters to write. Authority was based on society rather than what nature dictates, like Europe following Aristotle in the middle ages. A bit like following whatever Trump says and who asks him why about climate change or evolution or any of the other things he is an authority on? But then again social cohesion means they probably would not make such a person an authority so it balances out. Dmcq (talk) 14:22, 13 January 2017 (UTC)

see here. Count Iblis (talk) 20:21, 12 January 2017 (UTC)

I looked up the original quote in Littlewood's book to see if there was some context to make the meaning more clear, but there isn't really. Was Russel making a joke perhaps? I'm tempted to remove the quote unless someone can clarify what it means. --RDBury (talk) 05:18, 13 January 2017 (UTC)

I don't believe it was a joke and think the meaning is quite clear as I said above. The quote doesn't contribute much to the topic of the article though, it is more of a tidbit one might have at the end of some article like in a 'In popular culture' section. Dmcq (talk) 14:22, 13 January 2017 (UTC)

Chinese remainder theorem is broadly relevant. I'm not familiar with the details of the history, but I think it's sort of an a example of Dmcq's point. The Chinese had this figured out way before the Europeans did, and the very language and symbology we use to discuss it wasn't invented until... 1000+ years later. SemanticMantis (talk) 15:04, 13 January 2017 (UTC)

ratio of reduction for inscribed duals of duals of regular polyhedra?

Take a cube of volume 1 unit, inscribe an octahedron in the cube with corners of the octahedron at the centers of the faces and then inscribe a cube in the octahedron in the same way. If my calculations are correct, the inner cube should have volume 1/27. (If cube is +-1,+-1,+-1, so centers of the cube are at +-1,0,0 etc. then the centers of the faces of the octahedron are at +-1/3,+-1/3,+-1/3 giving a side 1/3 the length of the original). What is the equivalent fraction for starting with a Dodecahedron and inscribing a Icosahedron and then a Dodecahedron similarly?Naraht (talk) 21:50, 12 January 2017 (UTC)

I get the ratio of the sides of the dodecahedra is (2√5 + 5)/15, so the ratio of the volumes is that cubed. Some useful coordinates for a dodecahedron are (±φ, ±φ, ±φ), (±φ², 0, ±1), (±1, ±φ², 0), (0, ±1, ±φ²) where φ is the golden ratio. --RDBury (talk) 00:40, 13 January 2017 (UTC)

If the regular icosahedron and regular dodecahedron articles are accurate, then the ratio of the radii of the inscribed and circumscribed spheres for either of these solids is $\scriptstyle {\sqrt {\frac {5+2{\sqrt {5}}}{15}}}$. The quantity sought is the cube of the product of these two equal ratios, which is (in agreement with RDBury),

$\left({\frac {5+2{\sqrt {5}}}{15}}\right)^{3}={\frac {85+38{\sqrt {5}}}{675}}=0.251808...$

The ratio of the radii of the inscribed and circumscribed spheres for both the cube and the regular octahedron is $\scriptstyle {\frac {1}{\sqrt {3}}}$ (as found by Naraht). Since the regular tetrahedron is self-dual, this means that this ratio is the same for any Platonic solid and its dual - is there any insight that makes this result obvious? --catslash (talk) 01:45, 13 January 2017 (UTC)

Presumably the same holds for higher dimensional regular polytopes as well. I think what's going on becomes more apparent if you delve into exactly what is meant by a dual in the geometric sense. The dual of a convex polyhedron enclosing the origin and with vertices (a_i, b_i, c_i) is the polytope with faces a_ix + b_iy + c_iz = 1 and vice versa. The dual of the sphere with radius r and center at the origin is the sphere with radius 1/r and center at the origin. When you take duals of a polytope and a sphere in which it's inscribed, the dual sphere is inscribed in the dual polytope. If some polyhedron (regular or not) is inscribed in a sphere of radius p and a sphere if radius q is inscribed in it, then when you take duals you get spheres of radius 1/p and 1/q resp. If the ratio of the radii is p/q = λ, then the ratio of the radii of the duals is (1/q)/(1/p)=λ. I'm not sure if there are any non-regular polyhedra which would work; the inscribed and circumscribed spheres would not only have to exist but be concentric. Perhaps tetrahedra with vertices (±a, 0, b) and (0, ±a, −b). --RDBury (talk) 06:15, 13 January 2017 (UTC)

Actually there are other non-regular polyhedra with this property. One such is the "truncated octahedron" whose vertices are permutations of (±1, ±(√3-1), 0). It's not the standard truncated octahedron in that the hexagonal sides aren't regular but the sides next to squares are much longer than the sides between hexagons. Similarly there is a triangular prism with rectangles replacing the squares. --RDBury (talk) 22:13, 13 January 2017 (UTC)

Thank you. I was not familiar with polar reciprocation, so that was enlightening. --catslash (talk) 13:58, 16 January 2017 (UTC)