Thread: "higher dimensional book recommendations"

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I haven't read Jeffrey Weeks' book though he did coauthor my favorite
Scientific American article of all time titled "Is Space Finite?"

which is related to the work I've done cataloguing infinite regular
polyhedra .
Of particular interest to members of our list might be his familiar
games you can play in such tiled spaces
which includes a nice
implementation of chess in toroidal space. I managed to beat it, but
only because it plays deterministically. If you try that, be sure to
switch from the "fundamental domain" mode to "tiling" mode. We exchanged
a few emails a long time ago and I found him to be a very nice and
approachable guy.

Professor Coxeter is most
definitely a giant of mathematics. His book Regular Polytopes
is quite possibly
the definitive work on the subject. It's very dense reading but is a
great reference work to have around if only for the tables of 4D
vertices at the end. I consider it the bible of polyhedra. I was told
that he was a member of a polyhedra mailing list that I was on for
several years along with John Conway
, Mangus Wenninger
and others. He never
posted there but if he read the list then he probably read some of my posts.

I'm normally much more interested in scientist's works than I am of the
people themselves but I love a good personal story too and will have to
read Coxeter's. My favorite so far is The Man Who Loved Only Numbers:
The Story of Paul Erdos and the Search for Mathematical Truth
.
A really fun read about an amazing man.

-Melinda

Roice Nelson wrote:
> I've finished a couple books recently that I highly enjoyed and are
> apropos to the group.
>
> The Shape of Space
>
> by Jeffrey Weeks
>
> This does not require a deep math background - it is described as
> being at a high school level, but I really learned a ton and enjoyed
> it immensely. It is chock-full of dimensional analogy, interesting
> abstractions, and very fun to read with big, easy text and lots of
> pictures! It has also generated a number of thoughts for possible
> additional Rubik analogues in my mind. Briefly describing, the
> flexibility of topology opens up whole new worlds here, and if you
> abstract the original cube as just a 6-cell of faces on a topological
> sphere, all of a sudden there a veritable infinite number of new
> puzzles one could make. I've discussed possibly coding with my
> brother a 3D puzzle based on cell divisions of hexagons on a
> topological torus (e.g. a 12-cell is one option we did some sketches
> of; btw, the hexagonal tiling turns out to be important because 3
> cells still meet at each vertex). In the presentation we envision,
> the faces would have to stretch and deform when twisting due to the
> non-uniform curvature of a torus, but we hypercubists definitely don't
> care about such appearances on our screen ;)
>
> King of Infinite Space: Donald Coxter, the Man Who Saved Geometry
>
> by Siobhan Roberts
>
> This is a biography of Donald Coxeter, a new intellectual hero of mine
> after reading it. I really love the genre of mathematical/scientific
> biographies, and this is a good one. The book is much more history
> than math, with plenty of enjoyable anecdotal stories about Coxeter
> and his peers (Hardy, Einstein, Von Neumann, etc.). Overall it is an
> engaging, sweet portrait of someone enthralled with polytopes for his
> entire life.

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I haven't read Jeffrey Weeks' book though he did coauthor my favorite
Scientific American article of all time titled href="http://cosmos.phy.tufts.edu/%7Ezirbel/ast21/sciam/IsSpaceFinite.pdf">"Is
Space Finite?" which is related to the work I've done cataloguing href="http://www.superliminal.com/geometry/infinite/infinite.htm">infinite
regular polyhedra. Of particular interest to members of our list
might be his familiar
games you can play in such tiled spaces which includes a nice
implementation of chess in toroidal space. I managed to beat it, but
only because it plays deterministically. If you try that, be sure to
switch from the "fundamental domain" mode to "tiling" mode. We
exchanged a few emails a long time ago and I found him to be a very
nice and approachable guy.



Professor Coxeter is
most definitely a giant of mathematics. His book href="http://www.amazon.com/exec/obidos/ASIN/0486614808/">Regular
Polytopes is quite possibly the definitive work on the subject.
It's very dense reading but is a great reference work to have around if
only for the tables of 4D vertices at the end. I consider it the bible
of polyhedra. I was told that he was a member of a polyhedra mailing
list that I was on for several years along with href="http://en.wikipedia.org/wiki/John_Horton_Conway">John Conway,
Mangus Wenninger
and others. He never posted there but if he read the list then he
probably read some of my posts.



I'm normally much more interested in scientist's works than I am of the
people themselves but I love a good personal story too and will have to
read Coxeter's. My favorite so far is href="http://www.amazon.com/exec/obidos/ASIN/0786884061/qid=997660173/sr=1-1/ref=sc_b_1/107-3481249-3609337">The
Man Who Loved Only Numbers: The Story of Paul Erdos and the Search for
Mathematical Truth. A really fun read about an amazing man.



-Melinda



Roice Nelson wrote:
cite="mid:b5979e760807211614s3d75310eo9376a3fcc470e39@mail.gmail.com"
type="cite">

I haven't read Jeffrey Weeks' book though he did coauthor my favorite Scien=
tific American article of all time titled "Is Space Finite?" which is relat=
ed to the work I've done cataloguing infinite regular polyhedra. Of particu=
lar interest to members of our list might be his familiar games you can pla=
y in such tiled spaces which includes a nice implementation of chess in tor=
oidal space. I managed to beat it, but only because it plays deterministica=
lly. If you try that, be sure to switch from the "fundamental domain" mode =
to "tiling" mode. We exchanged a few emails a long time ago and I found him=
to be a very nice and approachable guy.

Professor Coxeter is most definitely a giant of mathematics. His book Regul=
ar Polytopes is quite possibly the definitive work on the subject. It's ver=
y dense reading but is a great reference work to have around if only for th=
e tables of 4D vertices at the end. I consider it the bible of polyhedra. I=
was told that he was a member of a polyhedra mailing list that I was on fo=
r several years along with John Conway, Mangus Wenninger and others. He nev=
er posted there but if he read the list then he probably read some of my po=
sts.

I'm normally much more interested in scientist's works than I am of the peo=
ple themselves but I love a good personal story too and will have to read C=
oxeter's. My favorite so far is The Man Who Loved Only Numbers: The Story o=
f Paul Erdos and the Search for Mathematical Truth. A really fun read about=
an amazing man.

-Melinda

Roice Nelson wrote:=20


I've finished a couple books recently that I highly enjoyed and are apropos=
to the group.
=A0
The Shape of Space by Jeffrey Weeks
=A0
This does not require a deep math background - it is described as being at =
a high school level, but I really learned a ton and enjoyed it immensely.=
=A0 It is chock-full of dimensional analogy, interesting abstractions, and =
very fun to read with big, easy text and lots of pictures!=A0 It has also g=
enerated a number of thoughts for possible additional Rubik analogues in my=
mind.=A0 Briefly describing, the flexibility of topology opens up whole ne=
w worlds here, and if you abstract the original cube as just a 6-cell of fa=
ces on a topological sphere, all of a sudden there a veritable infinite num=
ber of new puzzles one could make.=A0 I've discussed possibly coding with m=
y brother a 3D puzzle based on cell divisions of hexagons on a topological =
torus (e.g. a 12-cell is one option we did some sketches of; btw, the hexag=
onal tiling=A0turns out to be=A0important because 3 cells still meet at eac=
h vertex).=A0 In the presentation we envision, the faces would have to
stretch and deform when twisting due to the non-uniform curvature of a tor=
us, but we hypercubists definitely don't care about such appearances on our=
screen ;)
=A0
King of Infinite Space: Donald Coxter, the Man Who Saved Geometry by Siobha=
n Roberts
=A0
This is a biography of Donald Coxeter, a new intellectual hero of mine afte=
r reading it.=A0 I really love the genre of mathematical/ scientific biogra=
phies, and this is a good one.=A0 The book is much more history than math, =
with plenty of enjoyable anecdotal stories about Coxeter and his peers (Har=
dy, Einstein, Von Neumann, etc.).=A0 Overall it is an engaging, sweet portr=
ait of someone enthralled with polytopes for his entire life.=20














=20=20=20=20=20=20
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Hello All,   Roice, thanks for the book recommendations!  I also appreciate the info on V-Cubes.  I am going to get both the 6x6 and 7x7.  I would also love to have the entire collection, once they come out with cubes up to 11x11 in size.   Melinda, thanks for the Scientific American article, and for your work on infinite regular polyhedra and the related links!  I also prefer to read about the results of great mathematicians and scientists, but I too like to read about the people themselves.  I'll have to get these books eventually, but I have many books on my reading list right now, including both Hardy and Wright's classic An Introduction to the Theory of Numbers and the Feynman Lectures.   Roice and I have been in communication for some time now, and he recommended I update the group with my math results on Magic120Cell. I have updated my paper on the number of reachable positions of Magic120Cell which was originally a post on this group, and Roice has very kindly agreed to host it and my future papers on his website.  I have also solved the problem of the number of Magic120Cell programs with exactly/at most k colors.  These results can be found on my website,   http://mathproofs.bravehost.com/.   It occurred to me today that both Magic120Cell and the Megaminx can have any odd number of pieces per edge, and after I complete my papers on the nxnxn Rubik's Cube and the Magic120Cell coloring problem, I will try my hand at finding a formula for the number of different positions of a Magic120Cell program with any number of pieces per edge.   I hope my work on Magic120Cell will be interesting to some of you. The only book recommendation I have would be Nonplussed! by Julian Havil.  Only one chapter would be appropriate for this group, namely Hyperdimensions, Chapter 12.  It contains the most complete discussion of the n-dimensional sphere I have ever seen. It requires knowledge of calculus, but is well worth the effort!   All the Best,   David --- On Tue, 7/22/08, Melinda Green <melinda@superliminal.com> wrote: From: Melinda Green <melinda@superliminal.com> Subject: Re: [MC4D] higher dimensional book recommendations To: 4D_Cubing@yahoogroups.com Date: Tuesday, July 22, 2008, 1:45 AM I haven't read Jeffrey Weeks' book though he did coauthor my favorite Scientific American article of all time titled "Is Space Finite?" which is related to the work I've done cataloguing infinite regular polyhedra. Of particular interest to members of our list might be his familiar games you can play in such tiled spaces which includes a nice implementation of chess in toroidal space. I managed to beat it, but only because it plays deterministically. If you try that, be sure to switch from the "fundamental domain" mode to "tiling" mode. We exchanged a few emails a long time ago and I found him to be a very nice and approachable guy. href="http://en.wikipedia.org/wiki/Coxeter" target=_blank rel=nofollow>Professor Coxeter is most definitely a giant of mathematics. His book Regular Polytopes is quite possibly the definitive work on the subject. It's very dense reading but is a great reference work to have around if only for the tables of 4D vertices at the end. I consider it the bible of polyhedra. I was told that he was a member of a polyhedra mailing list that I was on for several years along with John Conway, Mangus Wenninger and others. He never posted there but if he read the list then he probably read some of my posts. I'm normally much more interested in scientist's works than I am of the people themselves but I love a good personal story too and will have to read Coxeter's. My favorite so far is The Man Who Loved Only Numbers: The Story of Paul Erdos and the Search for Mathematical Truth. A really fun read about an amazing man. -Melinda Roice Nelson wrote: I've finished a couple books recently that I highly enjoyed and are apropos to the group.   The Shape of Space by Jeffrey Weeks   This does not require a deep math background - it is described as being at a high school level, but I really learned a ton and enjoyed it immensely.  It is chock-full of dimensional analogy, interesting abstractions, and very fun to read with big, easy text and lots of pictures!  It has also generated a number of thoughts for possible additional Rubik analogues in my mind.  Briefly describing, the flexibility of topology opens up whole new worlds here, and if you abstract the original cube as just a 6-cell of faces on a topological sphere, all of a sudden there a veritable infinite number of new puzzles one could make.  I've discussed possibly coding with my brother a 3D puzzle based on cell divisions of hexagons on a topological torus (e.g. a 12-cell is one option we did some sketches of; btw, the hexagonal tiling turns out to be important because 3 cells still meet at each vertex).  In the presentation we envision, the faces would have to stretch and deform when twisting due to the non-uniform curvature of a torus, but we hypercubists definitely don't care about such appearances on our screen ;)   King of Infinite Space: Donald Coxter, the Man Who Saved Geometry by Siobhan Roberts   This is a biography of Donald Coxeter, a new intellectual hero of mine after reading it.  I really love the genre of mathematical/ scientific biographies, and this is a good one.  The book is much more history than math, with plenty of enjoyable anecdotal stories about Coxeter and his peers (Hardy, Einstein, Von Neumann, etc.).  Overall it is an engaging, sweet portrait of someone enthralled with polytopes for his entire life.

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Yeah, I really like the 7x7x7. It was pricey for sure (which is why I held
off on the 6^3), but it is quite an amazing product and works better than
the 5^3 I have. I've only solved it once so far and this first try took
hours!

On the topic of the
"hyper-Gigaminx"
and related puzzles, I wanted to mention that there is a corners only
variant of Megaminx (even number of pieces per side) called the
Impossiball.
It doesn't look so much like a Megaminx, but it is equivalent. I've never
handled one myself, but on the wiki page I saw that the twisting does
involve portions of the puzzle getting pushed outward because the slicing
doesn't happen on single planes. Maybe this is a little impure in a sense,
but since it is physically constructable, we are almost obliged to imagine a
4D version as well. And although I'm not aware of a 4-to-a-side Megaminx
having been constructed yet, maybe it is possible to think about other
even-sided versions of that and M120C too. I haven't thought enough about
it to be sure though.

Nelson and I were discussing some of these variants off-line and faked
pictures by messing with the settings. He named the 4D Impossiball
"Inconceivaball" :) Also, he surprised me with a picture of a 4D Pyraminx
Crystal...I
hadn't even known this 3D puzzle existed. It seems there are enough
permutation problems out there to keep one busy for a very long time! I
uploaded the screenshots Nelson and I did to the yahoo group photo area (click
here if
you'd like to see).

cya,
Roice

P.S. Thanks for info about the hypersphere chapter in Nonplussed, which is
now on my Amazon wishlist for that reason! Week's book has a chapter on the
hypersphere as well and I learned new things from it, but I also was aware
of it missing some discussion (he described great-spheres but did not go
into how it is possible to slice a hypersphere into two sections using
tori).


On 7/22/08, David Smith wrote:
>
> Hello All,
>
>
>
> Roice, thanks for the book recommendations! I also appreciate the info
>
> on V-Cubes. I am going to get both the 6x6 and 7x7. I would also love
>
> to have the entire collection, once they come out with cubes up to 11x11
>
> in size.
>
>
>
> Melinda, thanks for the Scientific American article, and for your work on
>
> infinite regular polyhedra and the related links! I also prefer
>
> to read about the results of great mathematicians and scientists, but I too
>
> like to read about the people themselves. I'll have to get these books
>
> eventually, but I have many books on my reading list right now, including
>
> both Hardy and Wright's classic *An Introduction to the Theory of Numbers*
>
> and the Feynman Lectures.
>
>
>
> Roice and I have been in communication for some time now, and he
>
> recommended I update the group with my math results on Magic120Cell.
>
> I have updated my paper on the number of reachable positions of
> Magic120Cell
>
> which was originally a post on this group, and Roice has very kindly agreed
>
> to host it and my future papers on his website. I have also solved the
>
> problem of the number of Magic120Cell programs with exactly/at most k
>
> colors. These results can be found on my website,
>
>
>
> http://mathproofs.bravehost.com/.
>
>
>
> It occurred to me today that both Magic120Cell and the Megaminx can
>
> have any odd number of pieces per edge, and after I complete my papers
>
> on the nxnxn Rubik's Cube and the Magic120Cell coloring problem,
>
> I will try my hand at finding a formula for the number of different
> positions
>
> of a Magic120Cell program with any number of pieces per edge.
>
>
>
> I hope my work on Magic120Cell will be interesting to some of you.
>
> The only book recommendation I have would be *Nonplussed!* by
>
> Julian Havil. Only one chapter would be appropriate for this group,
>
> namely Hyperdimensions, Chapter 12. It contains the most
>
> complete discussion of the n-dimensional sphere I have ever seen.
>
> It requires knowledge of calculus, but is well worth the effort!
>
>
>
> All the Best,
>
>
>
> David
>
> --- On *Tue, 7/22/08, Melinda Green * wrote:
>
> From: Melinda Green
> Subject: Re: [MC4D] higher dimensional book recommendations
> To: 4D_Cubing@yahoogroups.com
> Date: Tuesday, July 22, 2008, 1:45 AM
>
> I haven't read Jeffrey Weeks' book though he did coauthor my favorite
> Scientific American article of all time titled "Is Space Finite?"which is related to the work I've done
> cataloguing infinite regular polyhedra.
> Of particular interest to members of our list might be his familiar games
> you can play in such tiled spaceswhich includes a nice implementation of chess in toroidal space. I managed
> to beat it, but only because it plays deterministically. If you try that, be
> sure to switch from the "fundamental domain" mode to "tiling" mode. We
> exchanged a few emails a long time ago and I found him to be a very nice and
> approachable guy.
>
> Professor Coxeter is most
> definitely a giant of mathematics. His book Regular Polytopesis quite possibly the definitive work on the subject. It's very dense
> reading but is a great reference work to have around if only for the tables
> of 4D vertices at the end. I consider it the bible of polyhedra. I was told
> that he was a member of a polyhedra mailing list that I was on for several
> years along with John Conway,
> Mangus Wenninger and
> others. He never posted there but if he read the list then he probably read
> some of my posts.
>
> I'm normally much more interested in scientist's works than I am of the
> people themselves but I love a good personal story too and will have to read
> Coxeter's. My favorite so far is The Man Who Loved Only Numbers: The Story
> of Paul Erdos and the Search for Mathematical Truth.
> A really fun read about an amazing man.
>
> -Melinda
>
> Roice Nelson wrote:
>
> I've finished a couple books recently that I highly enjoyed and are
> apropos to the group.
>
> The Shape of Spaceby Jeffrey Weeks
>
> This does not require a deep math background - it is described as being at
> a high school level, but I really learned a ton and enjoyed it immensely.
> It is chock-full of dimensional analogy, interesting abstractions, and very
> fun to read with big, easy text and lots of pictures! It has also generated
> a number of thoughts for possible additional Rubik analogues in my mind.
> Briefly describing, the flexibility of topology opens up whole new worlds
> here, and if you abstract the original cube as just a 6-cell of faces on a
> topological sphere, all of a sudden there a veritable infinite number of new
> puzzles one could make. I've discussed possibly coding with my brother a 3D
> puzzle based on cell divisions of hexagons on a topological torus (e.g. a
> 12-cell is one option we did some sketches of; btw, the hexagonal
> tiling turns out to be important because 3 cells still meet at each
> vertex). In the presentation we envision, the faces would have to stretch
> and deform when twisting due to the non-uniform curvature of a torus, but we
> hypercubists definitely don't care about such appearances on our screen ;)
>
> King of Infinite Space: Donald Coxter, the Man Who Saved Geometryby Siobhan Roberts
>
> This is a biography of Donald Coxeter, a new intellectual hero of mine
> after reading it. I really love the genre of mathematical/ scientific
> biographies, and this is a good one. The book is much more history than
> math, with plenty of enjoyable anecdotal stories about Coxeter and his peers
> (Hardy, Einstein, Von Neumann, etc.). Overall it is an engaging, sweet
> portrait of someone enthralled with polytopes for his entire life.
>
>
>
>

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