THE ATTIC
INTERESTING AND USELESS SEQUENCES...
A few sequences completly useless...
Using Riemann's method of sums :
(1)
(2) (3)
Little personal sequence found by my buddy
David and I
(it is constructed by condering a circle where we inscribe verticle
trapeziums. By calculating the the area of those trapeziums in a
certain way, we stumble on the above sequence!)
Isolated sequence !
it's an archimedian sequence (polygons in a circle), prove later on!
Statistics/Counting
1) see Cesàro
2) c(n,k) triangles
3) Application of the law of large numbers
Let be a random sequence of n points in the square [0,1]x[0,1] and D_{n} the size of the set of these points in the circle of center 0 and radius 1, i.e. , then .
A bit of geometry
The volume of a sphere in *R*^{n}
is given by the formula:
where *n=2m*. Notice that this works
really well for *m=1*, *n=2* but it works as well for odd *n*
! In fact, for *n=1*, the length of the segment is *2R* so :
And that's no joke! Recall that Euler's
function check that for integer *n* *(n)=(n-1)!*
and more generaly for positif real *x*, *(x)=(x-1)(x-1)*
So according to the comment on the page about Euler and by condifering the function gamma as an
extension of the factorial function on *R*^{+}.
Try *n=3*, we do indeed get *4/3R*^{3} !
Similarly, for the surfaces, we can consider that in dimension *n*,
the volume of a sphere of radius R
can be given according to its surface by :
which gives us :
What is more funny, is that those fomulae
valid for all n can raide the
following question: Does the volume and surface of a sphere have a
maximum value for a given dimension ?
The answer seems to be yes in the sense that *V*_{n} and *S*_{n}
tend to *0* if *n* tends to infinity (yes, the gamma
function in n increase by a
factor of *(n-1)!* and so a lot more faster than the power of the
numerator).
By deriving the expresions of *V*_{n} and *S*_{n}
according to *n*, we find numericaly that the maximum value for
the surface is in *n=7,25695*... and for the volume in *n=5,25695*...
!
Therefor the sphere has a maximum value in dimension 5 and a maximum
surface in dimension 7!
Underneath are shown the graphs of volume and surface in dimension n of those "hypersphere", borrowed
from the fabulous encyclopedia
by Eric Weisstein.
Seriously, this is serious...
Fagnano and the complex numbers.
We can have eternal fun with complex numbers... Apparently a certain
Fagnano grant us the two following formulae :
Note anyway that by using the DL with z being complexe with the second
formula, we end up finding again :
which is no other than Leibniz's formula!
Pi and the euclidian spaces
During the external Cape maths challenge in *1994*,
dedicated to the study of the minimum radius of the disc from an
euclidian plane containing k
points to coordinates to integers, the candidates had to prove the
following result :
Let *D* be the disc center *z*_{0} and radius *r*
:
*D(z*_{0},r)={zC z-z_{0}r}
and *r*_{k}=min{r>0, z_{0}C, card(Z[i]D(z_{0},r))k}
therefore
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