More futuristic math... (4D)
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kirby
Sep 27, 2009, 3:53:44 AM9/27/09
to MathFuture
Just thought I'd continue some earlier threads.
The 4D geometry I've been talking about takes
the tetrahedron as topologically more minimal
than the cube (with good reason: it is) and therefore
switches to a different model of 3rd and 2nd powering,
introducing growing/shrinking tetrahedra and triangles
in place of cubes and squares.
Instead of XYZ (cubist), the matrix or scaffold is
developed from sphere packing in the well known
CCP pattern (same as FCC) with all edges the same
length.
In the CCP, every unit radius sphere is surrounded
by 12 others tangent to it, in a cuboctahedral
conformation. Successive layers of balls, packing
around a nuclear ball, define a progression of
1, 12, 42, 92, 162... balls, always in the same
cuboctahedral conformation. That's just how
CCP is defined.
Four unit radius balls, inter-tangent to one another,
define our unit of volume in this system, our model
of 3rd powering. The cube defined by two such
tetrahedra intersecting at mid-edges has a volume
of 3. The octahedron dual to this cube, the other
void in the CCP complementing the tetrahedra,
has a volume of 4. The rhombic dodecahedra
that encase each of the CCP balls in a space-
filling manner, such that edges between adjacent
ball centers penetrate their diamond faces at
90 degrees, have a volume of 6.
These easy whole number volumes in conjunction
with a sphere packing lattice sets the stage for a
bevy of geometric concepts, tightly organized and
accessible to grade schoolers, not just adults.
Linking the 4D tetrahedron to Karl Menger's
"geometry of lumps" provides a definitional context
that certifies this as a non-Euclidean geometry, but
not in the sense of jiggering with the fifth postulate,
although there's more we could say on that topic
as well.
All of this material was published in the 1970s by
a famous architect and later Medal of Freedom
winner. Since that time, students of this geometry
have spawned several new areas of investigation,
including elastic interval geometry which adds
dynamism to the edges. EIG was also inspired by
the work of Kenneth Snelson, the internationally
recognized sculptor who pioneered the tensegrity
genre. Gerald de Jong was an early developer of EIG.
You'll find several free tools on the Internet, such as
Tim Tyler's work at springie.com. Alan Ferguson's
SpringDance, a Delphi application, seems to be no
longer on-line.
The cuboctahedron defined by 12 balls around a
nuclear ball has a volume of 20 tetravolumes. We
also have a bridge to the five-fold symmetric
family and a modular system for dissecting these
shapes, including the A, B and T modules, all of
equal volume 1/24. Two As and 1 B combine to
give a space-filling irregular tetrahedron called
the MITE in our namespace, and depicted on
page 71 of Coxeter's 'Regular Polytopes'. There's
reason to bill this the minimum space-filler in
the sense that it's tetrahedral (simplest polyhedron)
and comes without the need for a complement.
The dissection into As and Bs is specific to this
4D geometry (4D in the sense of "four directional").
Given the links to architecture, art and computer
graphics ("geometry of lumps" makes sense in
ray tracing), it's not surprising that students looking
at careers in these areas are boning up on the
related syllabus, much of it free and on-line.
You'll find lots on YouTube as well.
The T module, also of volume 1/24, comes in
left and right handed versions, as do the A and B
modules. 120 of them (60 left and 60 right) define
a rhombic triacontahedron of volume 5. The
radius of this shape is just a tad less than
unity i.e. it almost shrink wraps the CCP ball.
Expanding this radius by the 3rd root of 3/2
takes the volume from 5 to 7.5, where the radius
turns out to be phi/sqrt(2). This larger rhombic
triacontahedron intersects the edges of the
volume 6 rhombic dodecahedron, which is
tangent to the CCP ball at its 12 diamond face
centers.
To make a little chart:
A, B, T vol 1/24
Mite (space-filling) vol 1/8
Tetrahedron vol 1
Cube vol 3
Octahedron vol 4
Rh Triacontahedron vol 5
Rh Dodecahedron vol 6
Rh Triacontahedron vol 7.5
Icosahedron vol ~18.51
Cuboctahedron vol 20
A few animations communicate this information
fairly succinctly. 'Clocktet' by Richard Hawkins,
which premiered at the Fuller Centennial in 1991
in Balboa Park in San Diego, was one of the first
in this genre. We've seen several since. Getting
more in the pipeline, from such shops as
Disney / Pixar, is a priority of my working group
in Portland. We also do a lot with Python,
showing students how to do their own ray tracings
and interactive geometry cartoons (POV-Ray and
VPython), plus how to use generators to yield such
sequences as 1, 12, 42, 92... (above), an easy
entre to programming, which by now has an
integral role in the digital math track we're
developing in the state of Oregon.
All of the above is well documented, including
write-ups of the various classroom pilots, open
source software, animations etc. There's an institute
(bfi.org) along with affiliated think tanks all
supporting one another in various ways. I've
personally been flown to Sweden and Lithuania
to brief my peers.
Universities have been slow to catch on however,
which is why the initiatives have been mostly
in the private sector. Dr. Arthur Loeb (MIT, Harvard)
helped us quite a bit. Dr. John Belt with SUNY has
also been of considerable assistance.
Fuller's chief collaborator on the magnum opus
in question, available on the web for some years,
was a career intelligence officer and author of
'Washington Itself', 'Cosmic Fishing' and 'Paradise
Mislaid'. I've included his picture on my Myspace
page.
Kirby Urner
myspace.com/4dstudios
The 4D geometry I've been talking about takes
the tetrahedron as topologically more minimal
than the cube (with good reason: it is) and therefore
switches to a different model of 3rd and 2nd powering,
introducing growing/shrinking tetrahedra and triangles
in place of cubes and squares.
Instead of XYZ (cubist), the matrix or scaffold is
developed from sphere packing in the well known
CCP pattern (same as FCC) with all edges the same
length.
In the CCP, every unit radius sphere is surrounded
by 12 others tangent to it, in a cuboctahedral
conformation. Successive layers of balls, packing
around a nuclear ball, define a progression of
1, 12, 42, 92, 162... balls, always in the same
cuboctahedral conformation. That's just how
CCP is defined.
Four unit radius balls, inter-tangent to one another,
define our unit of volume in this system, our model
of 3rd powering. The cube defined by two such
tetrahedra intersecting at mid-edges has a volume
of 3. The octahedron dual to this cube, the other
void in the CCP complementing the tetrahedra,
has a volume of 4. The rhombic dodecahedra
that encase each of the CCP balls in a space-
filling manner, such that edges between adjacent
ball centers penetrate their diamond faces at
90 degrees, have a volume of 6.
These easy whole number volumes in conjunction
with a sphere packing lattice sets the stage for a
bevy of geometric concepts, tightly organized and
accessible to grade schoolers, not just adults.
Linking the 4D tetrahedron to Karl Menger's
"geometry of lumps" provides a definitional context
that certifies this as a non-Euclidean geometry, but
not in the sense of jiggering with the fifth postulate,
although there's more we could say on that topic
as well.
All of this material was published in the 1970s by
a famous architect and later Medal of Freedom
winner. Since that time, students of this geometry
have spawned several new areas of investigation,
including elastic interval geometry which adds
dynamism to the edges. EIG was also inspired by
the work of Kenneth Snelson, the internationally
recognized sculptor who pioneered the tensegrity
genre. Gerald de Jong was an early developer of EIG.
You'll find several free tools on the Internet, such as
Tim Tyler's work at springie.com. Alan Ferguson's
SpringDance, a Delphi application, seems to be no
longer on-line.
The cuboctahedron defined by 12 balls around a
nuclear ball has a volume of 20 tetravolumes. We
also have a bridge to the five-fold symmetric
family and a modular system for dissecting these
shapes, including the A, B and T modules, all of
equal volume 1/24. Two As and 1 B combine to
give a space-filling irregular tetrahedron called
the MITE in our namespace, and depicted on
page 71 of Coxeter's 'Regular Polytopes'. There's
reason to bill this the minimum space-filler in
the sense that it's tetrahedral (simplest polyhedron)
and comes without the need for a complement.
The dissection into As and Bs is specific to this
4D geometry (4D in the sense of "four directional").
Given the links to architecture, art and computer
graphics ("geometry of lumps" makes sense in
ray tracing), it's not surprising that students looking
at careers in these areas are boning up on the
related syllabus, much of it free and on-line.
You'll find lots on YouTube as well.
The T module, also of volume 1/24, comes in
left and right handed versions, as do the A and B
modules. 120 of them (60 left and 60 right) define
a rhombic triacontahedron of volume 5. The
radius of this shape is just a tad less than
unity i.e. it almost shrink wraps the CCP ball.
Expanding this radius by the 3rd root of 3/2
takes the volume from 5 to 7.5, where the radius
turns out to be phi/sqrt(2). This larger rhombic
triacontahedron intersects the edges of the
volume 6 rhombic dodecahedron, which is
tangent to the CCP ball at its 12 diamond face
centers.
To make a little chart:
A, B, T vol 1/24
Mite (space-filling) vol 1/8
Tetrahedron vol 1
Cube vol 3
Octahedron vol 4
Rh Triacontahedron vol 5
Rh Dodecahedron vol 6
Rh Triacontahedron vol 7.5
Icosahedron vol ~18.51
Cuboctahedron vol 20
A few animations communicate this information
fairly succinctly. 'Clocktet' by Richard Hawkins,
which premiered at the Fuller Centennial in 1991
in Balboa Park in San Diego, was one of the first
in this genre. We've seen several since. Getting
more in the pipeline, from such shops as
Disney / Pixar, is a priority of my working group
in Portland. We also do a lot with Python,
showing students how to do their own ray tracings
and interactive geometry cartoons (POV-Ray and
VPython), plus how to use generators to yield such
sequences as 1, 12, 42, 92... (above), an easy
entre to programming, which by now has an
integral role in the digital math track we're
developing in the state of Oregon.
All of the above is well documented, including
write-ups of the various classroom pilots, open
source software, animations etc. There's an institute
(bfi.org) along with affiliated think tanks all
supporting one another in various ways. I've
personally been flown to Sweden and Lithuania
to brief my peers.
Universities have been slow to catch on however,
which is why the initiatives have been mostly
in the private sector. Dr. Arthur Loeb (MIT, Harvard)
helped us quite a bit. Dr. John Belt with SUNY has
also been of considerable assistance.
Fuller's chief collaborator on the magnum opus
in question, available on the web for some years,
was a career intelligence officer and author of
'Washington Itself', 'Cosmic Fishing' and 'Paradise
Mislaid'. I've included his picture on my Myspace
page.
Kirby Urner
myspace.com/4dstudios
kirby
Sep 29, 2009, 6:03:20 PM9/29/09
to MathFuture
I've blogged a copy of the post above (what I'm replying to) here:
http://worldgame.blogspot.com/2009/09/random-posting.html
Lots of hot links, for those wishing to do more homework on
this particular Future Math (except we encourage people to
say "Other Tomorrow" in place of "Future", to remind that
we have alternatives).
Kirby
On Sep 27, 12:53 am, kirby <kirby.ur...@gmail.com> wrote:
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