any spherical whiteboards out there?

[kirby urner]

My colleague Glenn and I were talking about school supply that we wonder if exits:  a spherical chalk or dry erase board. 

A lightweight dry erase ball about the size of a basket ball might be perfect. 

Glenn also pointed out there's "chalk board paint" available in hardware stores, so in theory one might start with a suitable sphere (an actual basket ball?) and paint it with a chalk-board-like surface. 

However something plastic, hollow, and more like a whiteboard sounds more practical.

A purpose of such a ball is to make graphs (in the sense of connected edges, networks) that close back on themselves, which would in many cases turn out to be polyhedrons (topologically). 

Four dots evenly spaced, with edges between them (six) would be a tetrahedron and so on.

Graph theory then defines the operation where we change marker color and put dots at the center of the areas and draw new edges connecting them, most typically crossing the previous edges and 90 degrees, typically bisecting them. 

That's to create the "dual graph" (dual polyhedron) -- we all know this of course.

If not too expensive, I'd give each student their own ball.  It'll come in useful again in global studies. :-D

Kirby

[Paul Libbrecht]

Kirby,

spherical blackboard (usually dark-greenish) are relatively common.
Should I inquire where I can find them?
They're not transparent (it's about sphere geometry, not space geometry).

Paul

kirby urner

2 March 2016 at 16:35

--
You received this message because you are subscribed to the Google Groups "MathFuture" group.
To unsubscribe from this group and stop receiving emails from it, send an email to mathfuture+...@googlegroups.com.
To post to this group, send email to mathf...@googlegroups.com.
Visit this group at https://groups.google.com/group/mathfuture.
For more options, visit https://groups.google.com/d/optout.

[kirby urner]

On Wed, Mar 2, 2016 at 8:28 AM, Paul Libbrecht <pa...@hoplahup.net> wrote:
Kirby,

spherical blackboard (usually dark-greenish) are relatively common.
Should I inquire where I can find them?
They're not transparent (it's about sphere geometry, not space geometry).

Paul

Thanks Paul!

I found these DIY instructions after posting the above:
http://www.instructables.com/id/Chalkboard-sphere/

(warning:  website is cluttered with ugly ads, really not pleasing)

One gets a whole kit with this setup, the Lénárt Sphere:
https://en.wikipedia.org/wiki/L%C3%A9n%C3%A1rt_sphere

Then there's this patent:
https://www.google.com/patents/US1207868

I can't quite figure out if this is what I'm looking for:
http://www.zazzle.com/the_crystal_ball_dry_erase_board-256564021534145977

(if that's about a spherical whiteboard, the presentation is extremely goofy)

One thing I want to show students is the Platonic Solids, a set, are closed under the operation of Doing the Dual ("dualing"?). 

Lets symbolize the two-way operation with <-->:

Tetrahedron <--> Tetrahedron

Cube <--> Octahedron

Dodecahedron <--> Icosahedron

Every element has an inverse and the op is associative so it's really a Group, as in a Group Theory group, but when do we see that in the literature?  Like there's this paper, but I'm not trying to be that complicated:

https://www.ocf.berkeley.edu/~wwu/papers/platonicsolids.pdf

Then there's this other operation where we *combine* duals to one another to form additional polyhedrons. 

Lets symbolize the combination operation with +:

Tetrahedron + Tetrahedron = Cube

Cube + Octahedron = Rhombic Dodecahedron

Dodecahedron + Icosahedron = Rhombic Triacontahedron

Then, starting with the new members, start taking the dual once again e.g. Rhombic Dodecahedron <--> Cuboctahedron.

Kirby

[kirby urner]

On Wed, Mar 2, 2016 at 8:47 AM, kirby urner <kirby...@gmail.com> wrote:

<< SNIP >>
 

One thing I want to show students is the Platonic Solids, a set, are closed under the operation of Doing the Dual ("dualing"?). 

Lets symbolize the two-way operation with <-->:

Tetrahedron <--> Tetrahedron

Cube <--> Octahedron

Dodecahedron <--> Icosahedron

Then of course in Martian Math we might pencil in these volumes:


Tetrahedron (1) <--> Tetrahedron (1)
Cube (3) <--> Octahedron (4)
Dodecahedron ((Φ² + 1) 3√2)  <--> Icosahedron (5 Φ² √2)

Cube + Octahedron = Rhombic Dodecahedron (6)
Dodecahedron + Icosahedron = Rhombic Triacontahedron (15√2 but shares vertexes with a Rhombic Dodeca at volume 7.5, also shrink-wraps the unit radius sphere when made of 120 so-called E-modules)

That's more rational whole number volumes than Earthlings are used to and I wouldn't expect many classrooms to take that radical step, except maybe in a global affairs class where we have to talk about history (about how the Martians invaded :-D).

Kirby

[kirby urner]

On Wed, Mar 2, 2016 at 9:17 AM, kirby urner <kirby...@gmail.com> wrote:


<< SNIP >>

Then of course in Martian Math we might pencil in these volumes:

Tetrahedron (1) <--> Tetrahedron (1)
Cube (3) <--> Octahedron (4)
Dodecahedron ((Φ² + 1) 3√2)  <--> Icosahedron (5 Φ² √2)
Cube + Octahedron = Rhombic Dodecahedron (6)
Dodecahedron + Icosahedron = Rhombic Triacontahedron (15√2 but shares vertexes with a Rhombic Dodeca at volume 7.5, also shrink-wraps the unit radius sphere when made of 120 so-called E-modules)

Oh yeah, then I forgot the Cuboctahedron, dual to the Rhombic Dodecahedron.

We tend to set that one's size using the same unit-radius balls used to make a tetrahedron:  twelve balls around a nuclear ball.  Volume 20.   

One may keep expanding outward with additional layers of balls, keeping the same cuboctahedron shape.  1, 12, 42, 92....  you know the sequence.

The sphere packing matrix is important and need not be divorced from all this Group Theory stuff.  That's how we do it in code school:

http://nbviewer.jupyter.org/github/4dsolutions/Python5/blob/master/STEM%20Mathematics.ipynb?flush_cache=true

A spherical whiteboard would be helpful in showing all this.  If we had as many as one per student then we could also bring them together for sphere packing demonstrations, same class (Martian Math -- or even Earthlings could do this if that brave :-D).

Kirby

[kirby urner]

On Wed, Mar 2, 2016 at 8:47 AM, kirby urner <kirby...@gmail.com> wrote:

 

Every element has an inverse and the op is associative so it's really a Group, as in a Group Theory group, but when do we see that in the literature?  Like there's this paper, but I'm not trying to be that complicated:

https://www.ocf.berkeley.edu/~wwu/papers/platonicsolids.pdf

To answer my own question, I'm guessing this example doesn't come up in the literature (Platonics as a Group under Duality) because Dual is a unary operator and Group Theory is all about binary operators.  Oh well.

So given the Platonics for a set, wherein each element P has a defined inverse and there's a unary operation, D, such that D(P) goes to ~P and vice versa, we have closure.  D(P) always gives another P.

That's a good segue to the *real* Group Theory then, which is based on permutations.  We'll get back to polyhedrons when we define the operation of "rotation".  

But rotation is a unary operator too isn't it?  So binary operators are *not* required....  OK, so maybe I'm a tad confused.

http://mathworld.wolfram.com/OctahedralGroup.html

Kirby

 

[jennifer kurtz]

Not exactly what you are looking for...  But, we used clear spherical ornaments (think Christmas tree) that you can open up.  We folded Platonic solid nets to fit inside of them.  We used dry erase markers on the outside/inside.  It was an inexpensive approach because I couldn't find what I really wanted--a huge clear sphere to teach with.  I wanted one that was already pre-made as was disappointed that I couldn't find one.

http://www.amazon.com/Naice-Christmas-Ornament-Plastic-Fillable/dp/B013I3E38U/ref=sr_1_1?ie=UTF8&qid=1456941358&sr=8-1&keywords=clear+sphere+ornament

I used the project in conjunction with Daud Sutton's book __Platonic and Archimedean Solids__ and Dan Radin's book and video on Platonic solids.

Not perfect but a cheap way for me to help the kids visualize what they were trying to understand.

Hope that helps.

Peace~
Jenn

---

From: kirby urner <kirby...@gmail.com>
To: mathf...@googlegroups.com
Sent: Wednesday, March 2, 2016 11:47 AM
Subject: Re: [Math Future] any spherical whiteboards out there?

Sent from Yahoo Mail

---

From: kirby urner <kirby...@gmail.com>
To: mathf...@googlegroups.com
Sent: Wednesday, March 2, 2016 11:47 AM
Subject: Re: [Math Future] any spherical whiteboards out there?

[kirby urner]

On Wed, Mar 2, 2016 at 10:03 AM, 'jennifer kurtz' via MathFuture <mathf...@googlegroups.com> wrote:

Not exactly what you are looking for...  But, we used clear spherical ornaments (think Christmas tree) that you can open up.  We folded Platonic solid nets to fit inside of them.  We used dry erase markers on the outside/inside.  It was an inexpensive approach because I couldn't find what I really wanted--a huge clear sphere to teach with.  I wanted one that was already pre-made as was disappointed that I couldn't find one.

http://www.amazon.com/Naice-Christmas-Ornament-Plastic-Fillable/dp/B013I3E38U/ref=sr_1_1?ie=UTF8&qid=1456941358&sr=8-1&keywords=clear+sphere+ornament

I used the project in conjunction with Daud Sutton's book __Platonic and Archimedean Solids__ and Dan Radin's book and video on Platonic solids.

Not perfect but a cheap way for me to help the kids visualize what they were trying to understand.

Hope that helps.

Peace~
Jenn

Thank you Jenn, that does help and I just ordered a 12-pack, along with one of these for testing with dry erase markers:

http://www.amazon.com/Darice-1105-89-Plastic-Ornament-140mm/dp/B002YGB0M0/

I like that Wooden Book series (Daud Sutton's is one).  I'm not so much a fan of Dan's stuff so far, but at least he covers the material.

Kirby

[John Bibby]

The Lenart sphere from Key Curriculum Press seems to be your thing!

See also

https://en.wikipedia.org/wiki/L%C3%A9n%C3%A1rt_sphere

Even if the spheres are no longer available, the book is well worth using. 

Non-Euclidean Adventures on the Lenart Sphere

JOHN BIBBY

==== You are invited to the following event in York that I am co-organising:

Borders and Beyond in the Middle East since 1914 ("BABITME") : the Middle East, WW1, etc. incl. local connections: (17-18 June 2016):

www.tinyURL.com/BABITME  

[John Bibby]

Dear Kirby

Re your query on graph theory & duality: you talk about 90 degrees and centres of areas. But what meaning can these have, as there is no metric in your representation? This is one of the dangers of such visual representations, useful though they can be in other respects. (With the Lenart Sphere book, of course the point is that there IS a metric, tho' it is non-Euclidean. Or, as Lenart puts it, a different set of axioms, esp. as regards the axion of parallels.)

Or have I misunderstood you?

JOHN BIBBY

==== You are invited to the following event in York that I am co-organising:

Borders and Beyond in the Middle East since 1914 ("BABITME") : the Middle East, WW1, etc. incl. local connections: (17-18 June 2016):

www.tinyURL.com/BABITME  

On 2 March 2016 at 15:35, kirby urner <kirby...@gmail.com> wrote:

[kirby urner]

On Wed, Mar 2, 2016 at 10:27 AM, John Bibby <johnbibby...@gmail.com> wrote:

The Lenart sphere from Key Curriculum Press seems to be your thing!

See also

https://en.wikipedia.org/wiki/L%C3%A9n%C3%A1rt_sphere

Even if the spheres are no longer available, the book is well worth using. 

Non-Euclidean Adventures on the Lenart Sphere

JOHN BIBBY

Thanks!

I've not seen one for under $100 yet.  E.G:

https://www.enasco.com/c/math/Geometry/L%26%23233%3Bn%26%23225%3Brt+Sphere%26%238482%3B/?ref=breadcrumb

I don't have an unlimited budget.

I'll keep my eye out for the book.

[ I'm not sure why spherical geometry is considered "non-Euclidean".  Sphere's are locally flat, like on the sandy beach where Euclidean proofs get made.  It's all one geometry in that sense, but then if you simply *eliminate* planes (perfect flatness), by definition... so much either/or in doing that. ]

Kirby

[kirby urner]

In Euclidean geometry with just the straight edge (unmarked) and compass, we have notions of "perpendicular" and "center of area".  No specific metric need apply, but then the compass itself supplies fixed width.  That's sufficient.

You're right though, when it comes to irregular triangles, we have several alternative ways to define "center":

http://mathworld.wolfram.com/TriangleCentroid.html

I might be doing non-Euclidean geometry in another sense:  I might not require my planes to be "infinite".  I'm into finite geometry.  I'm not hooked on the conventional use of "dimension" either as I like to treat everything as volumetric even if infra-tunably small i.e. of negligible / zero size.  That's a departure from most textbook definitions and I credit dimension theorist Karl Menger for it, among others.

Kirby

[jennifer kurtz]

Hope it works for you.  We did that project back when my son was 6 and he's 14 now ;-)  We actually wore out 2 copies of the Daud Sutton book and had to buy another copy.  Fun memories.  We need to revisit that activity!

He's doing geometry right now...not sure how I didn't think to do this with him again.  Thanks for helping to jog my memory.

Peace ~ Jenn

---

From: kirby urner <kirby...@gmail.com>
To: mathf...@googlegroups.com

Sent: Wednesday, March 2, 2016 1:16 PM

Subject: Re: [Math Future] any spherical whiteboards out there?

--
You received this message because you are subscribed to the Google Groups "MathFuture" group.
To unsubscribe from this group and stop receiving emails from it, send an email to mathfuture+...@googlegroups.com.
To post to this group, send email to mathf...@googlegroups.com.
Visit this group at https://groups.google.com/group/mathfuture.
For more options, visit https://groups.google.com/d/optout.

Sent from Yahoo Mail

---

From: kirby urner <kirby...@gmail.com>
To: mathf...@googlegroups.com

Sent: Wednesday, March 2, 2016 1:16 PM

Subject: Re: [Math Future] any spherical whiteboards out there?

[John Bibby]

I think you are not understanding the meaning of nonEuclidean. But anyhow the measure is irrelevant if what you are really interested in is graph theory or groups 

Sorry I am not an expert in any of these areas. I suggest you talk to someone you know who can explain the differences. But lenart's book is a good start!

JOHN

[kirby urner]

On Thu, Mar 3, 2016 at 1:21 AM, John Bibby <johnbibby...@gmail.com> wrote:

I think you are not understanding the meaning of nonEuclidean. But anyhow the measure is irrelevant if what you are really interested in is graph theory or groups 

nonEuclidean has to do with changes to Euclid's 5th axiom in conventional parlance. 
http://www.cut-the-knot.org/triangle/pythpar/Drama.shtml

Ordinary spherical trig, for example, is perfectly Euclidean so I could do hours of stuff with plastic balls and graphs thereon and never once mention Euclid's 5th axiom as being other than what Euclid used.  It's still the same geometry.

Euclid's geometry is not exclusively planar (per Book 13), only tends to be taught that way (with polyhedrons in the back of the book, most teachers never get to 'em).  He was deeply interested in polyhedrons.  He well understood the Earth was a ball, not any infinite plain.  http://www.veritas-ucsb.org/library/russell/FlatEarth.html

In less conventional parlance, one gets to "nonEuclidean" by other routes than jiggering with the 5th axiom, such as by altering the definitions. 

As someone interested in finite discrete (discontinuous) geometry, I have an interest in dispensing with "infinite planes" ("infinite lines" and so on) nor do I necessarily have a need for points of zero dimension, planes of two dimension and so on.  That's one language game, embedded on ordinary language by now, but not the only game in town.
 

In conventional Western Civ, the "perfect" and the "ideal" don't really exist but in Plato's World, an eternal place for Ideal Forms (the perfect circle and so on), a kind of Heaven for math objects.  If it's real, i.e. temporal / energetic, that's part of the fallen world (Christianity figures into it) where nothing's perfect by definition.  Westerners tend to disparage existence, trash the planet, and pray for a nice afterlife instead.

Given we have no "infinite planes of only two dimensions" in our everyday experience, maybe we could have a geometry that doesn't need those either, to get its work done?  It's a question worth asking.  Why should we care about lines being infinite when we never need them to be that long?

Karl Menger suggested awhile ago what he called a "geometry of lumps".  Points, lines and planes, shapes of any kind, are all volumetric in principle, not distinguished by dimension number.  We don't have height without width or breadth without height.  That's a different meaning of nonEuclidean.

But of course if you wanna go to college and impress your professors, you'll need to know how to think like a Roman and feign your faux greek. :-D  Teachers are not permitted to deviate from the State Curriculum too much!  I'm only hinting that it's possible to think outside the box.  Home schoolers have more freedoms perhaps.

Like, even though they're just axioms and definitions, like the rules for a board game, people treat them like proved dogmas, received wisdom, and like to use disembodied narrator voices to promulgate them to children, as if the gods themselves were Euclidean. 

If you say you have no need for infinite lines to your teacher, chances are you'll end up in the principal's office.  A number one lesson in math class is learning to keep your mouth shut if it goes against the received wisdom, right?

Sorry I am not an expert in any of these areas. I suggest you talk to someone you know who can explain the differences. But lenart's book is a good start!

JOHN

I think I'm sufficiently schooled in all this.  Hey, I went to Princeton, took honors calc from Thurston (the topologist) and blah blah.  I was a philosophy guy though which means I'm dangerous.  I feel free to take liberties (as in "liberal arts").

Kirby

[kirby urner]

On Thu, Mar 3, 2016 at 6:58 AM, kirby urner <kirby...@gmail.com> wrote:

Karl Menger suggested awhile ago what he called a "geometry of lumps".  Points, lines and planes, shapes of any kind, are all volumetric in principle, not distinguished by dimension number.  We don't have height without width or breadth without height.  That's a different meaning of nonEuclidean.

FYI, if interested in more philosophy:

http://coffeeshopsnet.blogspot.com/2009/03/res-extensa.html

Kirby

[kirby urner]

On Wed, Mar 2, 2016 at 5:41 PM, 'jennifer kurtz' via MathFuture <mathf...@googlegroups.com> wrote:

Hope it works for you.  We did that project back when my son was 6 and he's 14 now ;-)  We actually wore out 2 copies of the Daud Sutton book and had to buy another copy.  Fun memories.  We need to revisit that activity!

I just got the x12 70 mm balls, and x1 140 mm ball in the mail from Amazon.com!

They're transparent and easy to assemble / disassemble.  

Fill with colored sand?  To show volume relationships?  The sphere diameter will be the edge of our unit volume tetrahedron (picture four balls in a pile).

In Martian Math we have this sqrt(9/8) constant we sometimes apply (known as S3 in the jargon), given our D-edged unit of volume is a tad smaller than the unit cube's with edges R. 

==

... according to Wolfram Alpha (yes, I can also do the algebra myself):

https://flic.kr/p/EPg3Ag

... talking about sphere volume in Martian units.  Remember the rhombic dodecahedron of volume 6?  It's the space-filling casing for each ball (1, 12, 42, 92...).  This thing again:

http://nbviewer.jupyter.org/github/4dsolutions/Python5/blob/master/STEM%20Mathematics.ipynb

I haven't had time to do anything with them yet beyond throw together this photo-montage in my kitchen:

https://flic.kr/p/EoyeLX

Kirby

[jennifer kurtz]

I can't wait to see what you do with these.  I love the larger sized sphere.  I added it to my wish list. 

Peace~ Jenn

---

From: kirby urner <kirby...@gmail.com>
To: mathf...@googlegroups.com

Sent: Sunday, March 6, 2016 6:02 PM

Subject: Re: [Math Future] any spherical whiteboards out there?