Tag Archives: beauty

Huntington’s Red-Blue Set

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While reading Lillian Lieber’s book on infinity, I came across an astonishing example of infinite set (on pp. 207). Let’s call the property of existence of a rational number between given two rational number to be “beauty” (a random word introduced by me to make arguments clearer).

The set of rational numbers between 0 and 1 are arranged in ascending order of magnitude, and all of them are coloured blue. This is clearly a beautiful set. Then another another set of rational numbers between 0 and 1 is taken and arrange in ascending order of magnitude, but all of them are coloured red. This is also a beautiful set. Now, put these two sets together in such a way that each blue number is immediately followed by the corresponding red number. For example, 1/2 is immediately followed by 1/2 etc.  It appears that if we interlace two beautiful sets, the resulting set should be even more beautiful. But since each blue number has an immediate successor, namely the corresponding red number, so that between these two we can’t find even a single other rational number, red or blue, the resulting set is NOT beautiful.

The set created above is called Huntington’s Red-Blue set. It is an ingenious invention, where two beautiful sets combined together lead to loss of beauty. For more details, read the original paper:

Huntington, Edward V. “The Continuum as A Type of Order: An Exposition of the Modern Theory.” Annals of Mathematics, Second Series, 7, no. 1 (1905): 15-43. doi:10.2307/1967192.

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Beauty Beyond Language

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Mathematics is believed to be a language of symbols with  metamathematical meaning attached to them, for example:

(\forall \varepsilon>0) (\exists N \in \mathbb{N}) \ni m,n \geq N \Rightarrow |a_m - a_n| < \varepsilon

Can be translated in English as:

For every positive real number, \varepsilon, there exists a  natural number, N, such that, if  the natural numbers m and n are greater than or equal to N then absolute value of the difference between a_m and a_n is less than \varepsilon

Many contemporary mathematicians (big-shots) like Jean-Pierre Serre , believe that instead of logographic language (symbols represent the words themselves), we should use alphabetic language (words are made up of various letters) . This also makes sense to me, because as seen in above example, symbols seem to hide beautiful simplicity of a mathematical statement. But, on the other hand, alphabetic language is too lengthy to write.

Because of above debates about language of Mathematics, many mathematicians love Proof without Words, consider an example by Mariano Suárez-Alvarez  (http://mathoverflow.net/q/8847):

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1+2+\ldots + (n-1) = \frac{n(n-1)}{2}=\binom{n}{2}

But, there are some sub-domains in mathematics which doesn’t depend on language, for example Geometry. Let me illustrate this point with following Spanish video created by Cristóbal Vila (Instituto Universitario de Matemáticas y Aplicaciones of the Universidad de Zaragoza):


 

Irrespective of the language you speak, you can appreciate the relationship between different artistic works and mathematics (mainly, Geometry)

For full details of this project see:  http://www.etereaestudios.com/docs_html/arsqubica_htm/