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Conference rusure::math

Title:Mathematics at DEC
Moderator:RUSURE::EDP
Created:Mon Feb 03 1986
Last Modified:Fri Jun 06 1997
Last Successful Update:Fri Jun 06 1997
Number of topics:2083
Total number of notes:14613

1286.0. "M.I. Survey: 24 Theorems rated according to beauty" by NOEDGE::HERMAN (Franklin B. Herman DTN 291-0170 PDM1-1/J9) Tue Aug 14 1990 20:03

	David Wells conducted an informal survey back in the Fall 1988
    to the subscribers of the Mathematical Intelligencer (vol. 10, no. 4)
    asking them to evaluate 24 theorems on a scale of 0 to 10 for beauty.
    Approximately 90 people completed the survey. The theorems in 
    order of ranking are listed below. For more details consult
    Summer 1990 Mathematical Intelligencer (vol. 12, no. 3).

	Are any of your favorite theorems missing? Is there beauty
    in mathematics the same way one talks of beauty in the arts?
    I think so. Anyhow enjoy.

    -Franklin

--------------------------------------------------------------------------------

    1.	e^(i*PI) = 1

    2.	Euler's formula for a pohyedron : V + F = E + 2 

    3.	The number of primes is infinite.

    4.	There are 5 regular polyhedra.

    5.        1     1     1             PI^2  
         1 + --- + --- + --- + ... =   ------
             2^2   3^2   4^2             6

    6.	A continuous mapping of the closed disk into
	itself has a fixed point.

    7.	There is no rational number whose square is 2.

    8.	PI is transcendental.

    9.	Every plane map can be coloured with 4 colours.

    10.	Every prime of the form 4n + 1 is the sum of 
	two integral squares in exactly one way.

    11.	The order of subgroup divides the order of the group.

    12.	Any square matric satifyies its characteristic equation.

    13.	A regular icosahedron inscribed in a regular octahedron
	divides the edges of the Golden Ratio.

    14.       1             1             1               PI - 3
          ---------  -  ---------  +  ---------  - ... =  ------
          2 * 3 * 4     4 * 5 * 6     6 * 7 * 8              4

    15. If the points of the plane are each coloured red,yellow
	or blue, there is a pair of points of the same colour of
	mutual distance away.

    16.	The number of partitions of an integer into odd partitions
	is equal to the number of partitions into distinct partitions.

    17.	Every number greater than 77 is the sum of integers the sum
	of whose reciprocals is 1.

    18.	The number of representations of an odd number as the sum
	of 4 squares is 8 times the sum of its divisors, 24 times
	the sum of its odd divisors.

    19.	There is no equilateral triangle whose vertices are plane
	lattice points.

    20.	At any party, there is a pair of people who have the same
	number of friends present.

    21.	Write down the multiples of root2, ignoring fractional parts
	and underneath write the numbers missing from the first
	sequence:
	    1  2  4  5  7  8  9  11  12
	    3  6 10 13 17 20 23  27  30
	The difference is 2n in the n-th place.

    22.	The word problem for groups is unsolvable.

    23.	The maximum area of a quadrilateral with sides a,b,c,d
	is [(s-a)(s-b)(s-c)(s-d)]^(1/2), s is half the perimeter.

    24.	
	  5[(1 - x^5)(1 - x^10)(1 - x^15)...]^5    
        ----------------------------------------  =  
        [(1 - x)(1 - x^2)(1 - x^3)(1 - x^4)...]^6    

	     p(4) + p(9)x + p(14)x^2 + ..., 

	where p(n) is the number of partitions of n.

    
T.RTitleUserPersonal
Name
DateLines
1286.1"Name" that theoremRDVAX::NGTue Aug 14 1990 21:284
    Does each of these theorems has somebody's name attach to it?
    
    I am ashamed that I only know about half of them. The ones in
    questions are 10, 13-19, 21-24.
1286.2HPSTEK::XIAIn my beginning is my end.Wed Aug 15 1990 00:2064
>    1.	e^(i*PI) = 1

    Uh Franklin, I don' think this is right.

    I think this is known as Euler's identity.

>    2.	Euler's formula for a pohyedron : V + F = E + 2 

>    3.	The number of primes is infinite.

    No name for this, but was proved by Euclid.
   
>    4.	There are 5 regular polyhedra.

I think this is a direct result of 2. Hence, by Euler.

>    5.        1     1     1             PI^2  
>         1 + --- + --- + --- + ... =   ------
>             2^2   3^2   4^2             6

Also proved by Euler.

>    6.	A continuous mapping of the closed disk into
>	itself has a fixed point.

Brower's (sp?) fixed point theorem.

>    7.	There is no rational number whose square is 2.

Proved by some Greek guy of the Pathegorean school.

>    8.	PI is transcendental.

Lindmann's theorem

>    9.	Every plane map can be coloured with 4 colours.

The four color conjecture claimed to have been proven by the guys in
Illinois with a massive computer run.

>    10.	Every prime of the form 4n + 1 is the sum of 
>	        two integral squares in exactly one way.

Don't know about this one.

>    11.	The order of subgroup divides the order of the group.

Lagrange's theorem.  I am surprised that Sylow's theorem is not in 
the list.

Don't know the rest.

...

I am thinking about coming up with my list of favorate theorems, but 
can't do it...
Modern mathematics do not have beautiful easily understood, but somewhat 
isolated theorems any more.  Rather carving small individual sculptures,
modern mathematicians are more interested in laying fundations and 
building esoteric monuments.  In a sense, it was all David Hilbert's
fault.

Eugene
   
1286.3Sorry, 1. should be e^(i*PI) = -1NOEDGE::HERMANFranklin B. Herman DTN 291-0170 PDM1-1/J9Wed Aug 15 1990 12:0916
Re: .2:

>>    1.	e^(i*PI) = 1
>>
>>>>    Uh Franklin, I don' think this is right.
>>>>
>>>>    I think this is known as Euler's identity.

    Ooops!! Forgot the -1, i.e, 

         i*PI
	e     =  -1

    Thanks Eugene :^)

    -Franklin
1286.4more candidatesHERON::BUCHANANcombinatorial bomb squadWed Aug 15 1990 16:0934
1286.5Couple moreVMSDEV::HALLYBThe Smart Money was on GoliathThu Aug 16 1990 17:0327
1286.6a few more (it probably never stops)ALLVAX::JROTHIt's a bush recording...Thu Aug 16 1990 18:2838
1286.7GUESS::DERAMODan D'EramoFri Aug 17 1990 12:0815
	re .5,

>>    V(constant) = 0; V(x) = |a| + |b|; V(|x|) = |a| + |b| also; etc.

	Are you sure it's not V(x) = V(|x|) = b - a?  The sum looks
	like a sum of | delta x_n | which would be b - a.

>>    Also let N(y) = "the number of x for which f(x) = y"
>>    
>>    Clearly N(y) is only integer-valued.  Here a and b are implicit parameters.

	He must be ruling out functions like f(x) = 0 over [a,b] with a < b,
	for which N(0) has the cardinality of the continuum.

	Dan
1286.83 nitsHERON::BUCHANANcombinatorial bomb squadFri Aug 17 1990 13:1939
1286.9GUESS::DERAMODan D'EramoFri Aug 17 1990 15:469
	re .-1,

>>	>	Are you sure it's not V(x) = V(|x|) = b - a?
>>
>>		|b-a|

	That's what I said :-) ... the context was functions on [a,b].

	Dan
1286.10GUESS::DERAMODan D'EramoFri Aug 17 1990 15:5727
1286.11How does this look, nice person?VMSDEV::HALLYBThe Smart Money was on GoliathFri Aug 17 1990 16:5925
>>>	>	Are you sure it's not V(x) = V(|x|) = b - a?
>>>
>>>		|b-a|
    
    No.  Consider the case [-2,3].  I'll just plot the lattice points:
    
                   x				|x|
    
     3                *                              *
     2               *                          *   *
     1              *                            * *
     0             *                              *
    -1            *
    -2           *
    
                   0                              0
    
    In both cases the variation is 5.  You can see that the integral of
    N(y) dy is also 5.
    
    Note that N(y) is the number of y's for which f(x) = y.  Thus there
    is no "continuum problem" -- you'd need a vertical line, hence not a
    (continuous or otherwise) function, to have that big a value for N(y).
    
      John
1286.12My favourite...UTRUST::DEHARTOGmoduladaplisprologopsimulalgolSat Aug 18 1990 19:436
What puzzled me since my teacher at highschool told me was the fact that

	"you can't divide an angle in three equal angles with ruler and
	 compasses".

Or is there a more general theorem behind this?
1286.13HPSTEK::XIAIn my beginning is my end.Sat Aug 18 1990 20:095
    re .12,
    
    Yep.
    
    Eugene
1286.14O(Reals) > O(Rationals)CHOVAX::YOUNGTurf = Ownership - AccountabilitySun Aug 19 1990 05:078
    One thing that is unclear here, is whether it is the *Theorem* or the
    *Proof* of that theorem that is being considered.
    
    For instance I find the "Countability" theorem itself to be fairly
    plain.  But its proof is, to me, one of the most beautiful work in all
    of mathematics.
    
    --  Barry
1286.15rat-warren alertHERON::BUCHANANcombinatorial bomb squadSun Aug 19 1990 12:54107
	Arrgh!   There is a rapidly-increasing number of discussions, of
various degrees of gravity, going on here.   Let me try and simplify them.

-------------------------------------------------------------------------------

Re my .8, I received mail from Dan saying that there was an interesting
ambiguity in the following:

>	With swaps permitted, any number above 23 can be represented.
>	With swaps and negative numbers permitted, any number can be 
>	represented, trivially.
>	With negative numbers, but no swaps, I don't know which of the
>	numbers -% to 77 now become possible: this is an interesting puzzle.

	It concerns "swap".   I meant "swap" to mean "duplicate copy", *not*
"transposition".   In English English at least, this usage is OK:  it derives 
from bubblegum-card collecting, or stamp collecting.   If I have n copies of the
same card, then n-1 of them are called "swaps" because I am prepared to swap
them with some other collector, for new cards that I don't have yet!

	I guess to us computer weenies, the sense "transposition" should be
the first one to come to mind, because of the phrase "swapped out", but on
this occasion, for me, it was the other sense which occurred to me when I
picked the word.

	The ambiguity derives special force because of the juxtaposition
of this discussionette with the sum-of-squares mini-conversazione, where
the duplicate-vs-transposition issue is at the heart of counting the
possibilities.

	What I was saying on these two problems is still true, I believe.
Status: semantic nit dealt with, interesting puzzle is still open.

-------------------------------------------------------------------------------

Re: Dan's .9
>>>	>	Are you sure it's not V(x) = V(|x|) = b - a?
>>>
>>>		|b-a|
>
>	That's what I said :-) ... the context was functions on [a,b].

	Yes, of course, I wasn't thinking straight.
But on this same topic, Re John's .11:

>>>>	>	Are you sure it's not V(x) = V(|x|) = b - a?
>>>>
>>>>		|b-a|
>    
>    No.  Consider the case [-2,3].
>    
>    In both cases the variation is 5.  You can see that the integral of
>    N(y) dy is also 5.
>    
>    Note that N(y) is the number of y's for which f(x) = y.  Thus there
>    is no "continuum problem" -- you'd need a vertical line, hence not a
>    (continuous or otherwise) function, to have that big a value for N(y).
>    
>      John

	3 - (-2) *is* 5, John, so you in fact agree with us!

Status:  trivial errors hopefully removed, does anyone have anything 
substantive to add to John's intro & Jim's reply?

-------------------------------------------------------------------------------

>Re: Dan's .10:
>
>	I'm not too familiar with this theorem, though I read about it
>	once in a number theory book (Hardy and Wright?).  The
>	"representations" being counted must be ordered quadruples of
>	signed integers in order to get eight solutions for one:
>	<-1,0,0,0>,<1,0,0,0>,<0,-1,0,0>,<0,1,0,0>,....  It is supposed
>	to be true of all numbers, not just of all odd numbers.  The
>	24x part must be about even numbers.

	...and the 8x part must be about *odd* numbers (try n=4).
Status: so we probably know what the Theorem must have been now.


Re: Hans' .12

>What puzzled me since my teacher at highschool told me was the fact that
>
>	"you can't divide an angle in three equal angles with ruler and
>	 compasses".
>
>Or is there a more general theorem behind this?

The Theory of Field Extensions is treated in some detail elsewhere in this
Notesfile (try dir /title=Galois).
Status: referenced to another Note.

-------------------------------------------------------------------------------

Re: Barry's .14

>    One thing that is unclear here, is whether it is the *Theorem* or the
>   *Proof* of that theorem that is being considered.
    
	Yes, but this is too important a point to make in a 
recreational topic.   :-).

-------------------------------------------------------------------------------

Moral: it is tricky to have 24 simultaneous discussions in the same topic.
1286.16GUESS::DERAMODan D'EramoMon Aug 20 1990 02:0713
        Rereading .0, the subscribers were asked to rank 24
        specific theorems, and produced the ordering shown in .0. 
        The subscribers weren't asked to name "their" most
        "beautiful" theorems.
        
        If I had put together the list, there would be more of a
        slant towards topology, set theory, and perhaps number
        theory. For example: Urysohn's theorem and Tychonoff's
        theorem in topology, Zermelo's Well Ordering theorem.
        
        There should also be a list for favorite conjectures.
        
        Dan
1286.17ALLVAX::JROTHIt's a bush recording...Mon Aug 20 1990 16:0114
    This is another instance of "see it once, see it again..."

    I was looking at the current issue of the mathematical monthly
    and Paul Halmos lists about a dozen major 20'th century ideas
    in mathematics.

    The list is pretty uneven - amongst truly impenetrable concepts
    (like K-theory) there are simple applied things like the FFT.

    There was stuff like deBranges proof of the Bieberbach conjecture,
    the classification of finite simple groups, the 4 color map theorem,
    chaos theory, fractals, etc.

    - Jim
1286.18A (very) little helpCIVAGE::LYNNLynn Yarbrough @WNP DTN 427-5663Mon Aug 27 1990 16:4319
>    23. The maximum area of a quadrilateral with sides a,b,c,d
>	is [(s-a)(s-b)(s-c)(s-d)]^(1/2), s is half the perimeter.
If any of a,b,c,d is zero, this reduces to Hero[n]'s formula for the area 
of any triangle. It is also the formula for the area of a quadrilateral 
inscribed in a circle. Does there exist a similar formula for the area of a 
convex pentagon?

>    24.	
>	  5[(1 - x^5)(1 - x^10)(1 - x^15)...]^5    
>        ----------------------------------------  =  
>        [(1 - x)(1 - x^2)(1 - x^3)(1 - x^4)...]^6    
>
>	     p(4) + p(9)x + p(14)x^2 + ..., 
>
>	where p(n) is the number of partitions of n.

I'm pretty sure this is Ramanujan's. To paraphrase (that is, misquote)
Hardy's comment on this and similar theorems, "It must be correct. No one
in his right mind would have conceived of it if it weren't."