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#1 2008-05-11 02:35:41

John E. Franklin
Member
Registered: 2005-08-29
Posts: 3,562

Venn Diagram with 4 spheres in tetrahedral fashion

You can make a 4 variable
venn diagram by
placing 4 spheres at the
corners of a tetrahedron.
Make the spheres overlap
just like the 3 circle Venn
diagrams you've seen
on flat paper.


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#2 2008-05-11 04:06:15

John E. Franklin
Member
Registered: 2005-08-29
Posts: 3,562

Re: Venn Diagram with 4 spheres in tetrahedral fashion

Here's my drawing for it.
Click image to make big.

View Image: TetraVen.jpG

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#3 2008-05-12 09:09:05

John E. Franklin
Member
Registered: 2005-08-29
Posts: 3,562

Re: Venn Diagram with 4 spheres in tetrahedral fashion

So there are 16 places in the 4-ball Venn Diagram.
The places look like the parts of a pyramid, but
not the pyramids with a square base like the Egyptians,
but instead a pyramid with a triangle for the bottom.
1. the four corners called vertices
2.   "
3.   "
4.   "
5. the six edges of the pyramid
6.   "
7.   "
8.   "
9.   "
10.  "
11.  the four triangular sides called faces
12.  "
13.  "
14.  "
15.  the space inside the pyramid
16.  the space outside the pyramid

These 16 places are like the
16 vertices of a 4-D hypercube,
which is more symmetrical, but
unfortunately we don't live in
4-D, so we can't see the
perfect symmetry of that, but
it would be like the differenence
between a square to a cube, it
gets twice as many corners as
a cube.
But this thread is really about
the Venn diagram, since you
can see it in 3-D, which is nice.
But the 16 places of the tetrahedron
are not uniform between all 16, because
they form groups, such as
sides, edges, corners, inside and outside!!

Last edited by John E. Franklin (2008-05-12 09:11:28)


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#4 2008-05-13 16:00:24

John E. Franklin
Member
Registered: 2005-08-29
Posts: 3,562

Re: Venn Diagram with 4 spheres in tetrahedral fashion

Venn diagrams and boolean equations go hand-in-hand.
Another construction is the Karnaugh map, which is
less well known to the mathematician, as it was designed
for digital electronic designers.  And don't forget the
hypercubes, as they are actually the same thing as a
Karnaugh map, except even better due to perfect symmetry.
If you would like to see a paper I wrote on this subject,
which is perhaps not very well written yet, follow this
link and X-out the silly ads overlayed on top of my paper.
http://johnericfranklin.250free.com
or same:
http://johnericfranklin.250free.com

This long webpage might give you some insight as to where
I'm coming from, or it might might bring more confusion to
the table!! (quite likely, lol)


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#5 2009-04-07 06:58:56

John E. Franklin
Member
Registered: 2005-08-29
Posts: 3,562

Re: Venn Diagram with 4 spheres in tetrahedral fashion

Here's a 4-D hypercube I just drew that is
unlike others I have seen.  It makes use
of the idea that tetrahedrons fit nicely
inside cubes.  The green and the red dots
are not really connected.  They look like
they are in the picture just because of the
overall cube shape I started with.  To travel
from a red to a green dot would require
travelling first through either a yellow,
purple, or blue dot.

View Image: JohnsHypercube.pnG

Last edited by John E. Franklin (2009-04-07 08:06:02)


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#6 2010-12-13 06:15:31

John E. Franklin
Member
Registered: 2005-08-29
Posts: 3,562

Re: Venn Diagram with 4 spheres in tetrahedral fashion

Glad to hear it!!!


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