Conference rusure::math

    You wouldn't have the Betti numbers perchance?

    - Jim

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The time-of-day you entered this note makes sense: its typically the time
the one sees weird shapes...
If I remember correctly:
Normal (Euclidian?) objects have the following property:

 vertices + faces = edges + 2

Does you object still have straight edges and flat faces?

Re .1 What are Betti-numbers?

Re .3
Sorry Andrew, I didn't realize that your time of day is not necessarely
the same as the time-of-day at the notesfile-location.
								Hans.

        Andrew,
        
        Give them the first half (only) of that hint.
        
        Dan

    Well obviously this is a seasonal one. I think you remember the craze a
    while back for keeping knives and razor blades sharp by placing a
    pyramid over them. Did you think that the mystic powers of a pyramid
    would stop the needles falling off your christmas tree if you placed a
    pyramid over them.
    
    Anyway, the construction is as follows:
    
    Take a tetrahedron (T) and cut a tetrahedral hole out of it. The base of
    the hole lies in the base of T and the apex of the hole is in one of
    the other faces of T.
    
    Myself, I think I'll just water the tree.
    
    dick
    
    Sorry, I almost forgot:-    Seasonal [:-)]'s

	Re: various replies.

	Betti numbers, I seem to remember, are what you get when you
take a structure in some number of dimensions, triangulate it, and
do some quotienting to derive expressing for the number of holes,
twists and things in the structure.   I'm hazy on the details.   If
J.Roth would care to describe the algorithm, I would perform it over
this solid, and give you the results.   Fair enough.

	The f+v=e+2 thing is the Euler criterion (hence the title, btw
I pronounce "Euler" as "oiler").   This doesn't hold, so clearly 
something funny is going on.   But not very funny.

	All the faces are flat.   An edge only exists where two faces
meet (each of the 2 faces being in a different plane).   A vertex only
exists where three or more edges meet, (no three of these edges being
in the same plane).

	I don't think I'll include the clue I gave in the first version
of this puzzle, which led Dan straight to the answer, until I see some
evidence that people have actually been tossing around some shapes in
their mind.

Regards,
Andrew.

	.6 was put in while I was editing .7.   It's basically correct,
but if you cut a pyramidical hole out of one side of a pyramid, then
there no need for the apex of the hole to be in one of the other faces
of T.

	So it's the 'hexagonal' face, the one with the hole in it, which
screws up Euler.

	To respond to the seasonal note, here at least Christmas tree
sellers dip the roots of each tree they sell in boiling water, to ensure
that the tree can only be used for one year.   Sad, huh?

Regards,
Andrew.

see notes 214 & 215.

    about Euler, I just got 2 books "Introduction to analysis of the
    infinite" by Euler.
    this is the new English translation of his work, published by
    Springer-Verlag, the 2 books cost $110 
    isbn 0-387-96824-5 , call QA295.E8413, and isbn 0-387-97132-7 
    
    the contents are:
    Book I:
    1. On fractions in general
    2. on the transformation of functions
    3. on the transformation of functions by substitution
    4. on the development of functions of infinite series
    5. concerning functions of two or more variables
    6. on Exponentials and logarithms
    7. exponentials and logarithms expressed through series
    8. on transcendental quantities which arise from the circle
    9. on trinomial factors
    10. on the use of the discovered factors to sum infinite series
    11. on other infinite expressions for arcs and sines
    12. on the development of real rational functions
    13. on recurrent series
    14. on the multiplication and division of angles
    15. on series which arise from products
    16. on the partition of numbers
    17. using recurrent series to find roots of equations
    18. on continued fractions

    Book II:
    1. on curves in general
    2. on the change of coordinates
    3. on the classification of algebraic curves by orders
    4. on the special properties of lines of any order
    5. on the second order lines
    6. on the subdivision of second order lines into genera
    7. on the investigation of branches which go to infinity
    8. concerning asymptotes
    9. on the species of third order lines
    10. on the principle properties of third order lines
    11. on fourth order lines
    12. on the investigation of the configuration of curves
    13. on the disposition of curves
    14. on the curvature of curves
    15. concerning curves with one or several diameters
    16. on finding curves from the properties of the ordinates
    17. on finding curves from other properties
    18. on the similarities and affinities of curves
    19. on the construction of equations
    20. on Transcendental curves
    21. on the solution to several problems pertaining to the circle
    Appendix:
    1. on the surface of solids
    2. in the intersection of surface and arbitrary plane
    3. on sections of cylinders, cones, and spheres
    4. on the change of coordinates
    5. on second order surfaces
    6. on the intersection of two surfaces

    /Nasser
    I spell checked