Help!! (Sage)

[David Chandler]

I think there is some Sage expertise in this forum.  I need some Sage advice.

I am trying to do a unit on geometric transformations using matrices.  The polygon vertices are the column vectors of a matrix.  I want to hit the polygon repeatedly with a transformation matrix (for instance one that does a rotation combined with a dilation), then I want to plot the original polygon and the sequence of transformed polygons all on the same plot.  (The particular case I am describing should produce a spiral pattern.)

I can see how to create matrices and transform them in Sage, and I can see how to create polygons, and plot them, but I don't see how to create matrices that represent the polygon or transform a polygon with a matrix or convert a polygon to a vertex matrix or convert a matrix to a polygon.  Plotting the pre-image and the image polygons all on the same graph is another issue.

--David Chandler

[michel paul]

See if this helps. I was investigating these kinds of things awhile back.

I've only used a rotation matrix here, but it would of course be possible to add other transformations in this kind of scheme.

To plot pre-image and image together, add them in a show() function.

- Michel

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--

===================================
"What I cannot create, I do not understand."

- Richard Feynman

===================================
"Computer science is the new mathematics."

- Dr. Christos Papadimitriou
===================================

[David Chandler]

Hi Michel,

Yes!  That does the trick.  Thanks.

The key step I was lacking was how to get a list of points back out of the matrix format.  Apparently <matrix>.columns() creates a list of points that can then be used in the polygon construction.  So the strategy is to maintain a list of points for the most recently transformed polygon.  Whenever I need to transform the list I use matrix() to turn it into a matrix with the point pairs acting as column vectors.  Once I have transformed the matrix, I can extract the new list of points using the columns() method.

The other item I improvised, since I wanted to "show" a list of polygons generated in a loop is based on your  show(p1 + p2 + p3).  I create a running "sum" of polygons using 

     polysum+=polygon(p)

That actually works.  I don't know how to visualize the structure of polysum except to see it as a sum or concatenation of a series of polygons.

--David Chandler

[David Chandler]

By the way, how did you share your sage notebook with me?  Is that what happens when you "publish" the notebook?  I tried using my downloaded version of Sage and it published it to localhost.  I assume that's on my computer.

--David Chandler

[David Chandler]

I saved my worksheet to a sws file and am attaching it here.

--David Chandler

[michel paul]

I really like that spiral of triangles! Yes, very nice. 

The way I shared my stuff through a simple link was via the single cell server. It is so handy! Just throw an idea down and send a link. You only have one cell; however, you can actually do a lot with one cell. Especially if you're going to set up an interactive window. The user doesn't necessarily have to understand Sage, but they can still use the interact, and theoretically they could study the code and modify to see what happens if they wanted to. They wouldn't break anything, as the link always goes to the original.

The public servers no longer support publishing, unfortunately. It used to be that if you had an account on sagenb.org, I'd be able to either publish a file that you could access, or I'd be able to share a file directly with you. However, all that got shut down, I think because once you've shared a file with someone they can't delete it - the sharer has to, and I think that might have been abused. Just speculating, but all I know is that at one point the public Sage servers were all down for some time, and when they went back up the publishing and sharing features were off with some mention of abuse having occurred. 

However, this single cell server got created, and I have recently been finding it a wonderful way to share ideas. For example, here's another one I recently shared with some people regarding polar plotting. The user can change the polar expression and the boundaries to explore how the curve unfolds in different intervals. You can share these kinds of things with anyone, even if they don't have a Sage account. I think the implications of that are fantastic. In trying to share things with my colleagues, I'd often experience resistance in that they didn't want to have to bother dealing with a Sage account (even though it's free). Now they have no excuse! : )

- Michel

[michel paul]

One more thing - If you'd like to create permanent text in your files, not just using comments in the computational cells, you can open up text processing cells:

• Let the cursor hover above or below an already existing computational cell.
• When you see a blue line appear, shift click it.
• You should see a text processing cell open! It understands LaTeX, so you can create publishable quality documents.
• It also handles graphics and links.

- Michel

[David Chandler]

OK, here is a permalink using Sage Cell Server.  How long is the temporary link good for?

On Wed, Apr 3, 2013 at 7:29 AM, michel paul <python...@gmail.com> wrote:

[David Chandler]

Mission accomplished!

With the critical item of input you gave me, I have now generalized the tool to display results of iterated composite x-reflections, y-reflections, dilations, rotations, and translations.

See it here.

Thanks.

[sin...@optonline.net]

Thank you all, as a former system engineer and current math teacher looking to integrate free as well as open source tools into the classroom I routinely learn a great deal from this group!

On Wed, Apr 3, 2013 at 6:03 PM, David Chandler <david...@gmail.com> wrote:
Mission accomplished!
With the critical item of input you gave me, I have now generalized
the tool to display results of iterated composite x-reflections,
y-reflections, dilations, rotations, and translations.

See it here [1].
Thanks.
On Wed, Apr 3, 2013 at 11:44 AM, David Chandler wrote:
OK, here is a permalink [3] using Sage Cell Server.  How long is

the temporary link good for?

On Wed, Apr 3, 2013 at 7:29 AM, michel paul wrote:
I really like that spiral of triangles! Yes, very nice. 
The way I shared my stuff through a simple link was via the single

cell server [5]. It is so handy! Just throw an idea down and send a

link. You only have one cell; however, you can actually do a lot with
one cell. Especially if you're going to set up an interactive
window. The user doesn't necessarily have to understand Sage, but
they can still use the interact, and theoretically they could study
the code and modify to see what happens if they wanted to. They
wouldn't break anything, as the link always goes to the original.
The public servers no longer support publishing, unfortunately. It

used to be that if you had an account on sagenb.org [6], I'd be

able to either publish a file that you could access, or I'd be
able to share a file directly with you. However, all that got shut
down, I think because once you've shared a file with someone they
can't delete it - the sharer has to, and I think that might have
been abused. Just speculating, but all I know is that at one point the
public Sage servers were all down for some time, and when they went
back up the publishing and sharing features were off with some mention
of abuse having occurred. 
However, this single cell server got created, and I have recently been
finding it a wonderful way to share ideas. For example, here's

another one [7] I recently shared with some people regarding polar

plotting. The user can change the polar expression and the boundaries
to explore how the curve unfolds in different intervals. You can share
these kinds of things with anyone, even if they don't have a Sage
account. I think the implications of that are fantastic. In trying to
share things with my colleagues, I'd often experience resistance
in that they didn't want to have to bother dealing with a Sage
account (even though it's free). Now they have no excuse! : )
- Michel

On Tue, Apr 2, 2013 at 11:48 PM, David Chandler wrote:
I saved my worksheet to a sws file and am attaching it here.
--David Chandler

On Tue, Apr 2, 2013 at 11:10 PM, David Chandler wrote:
By the way, how did you share your sage notebook with me?  Is that
what happens when you "publish" the notebook?  I tried using my
downloaded version of Sage and it published it to localhost.  I
assume that's on my computer.
--David Chandler

On Tue, Apr 2, 2013 at 11:07 PM, David Chandler wrote:
Hi Michel,
Yes!  That does the trick.  Thanks.
The key step I was lacking was how to get a list of points back out of

the matrix format.  Apparently .columns() creates a list of points

that can then be used in the polygon construction.  So the strategy
is to maintain a list of points for the most recently transformed
polygon.  Whenever I need to transform the list I use matrix() to
turn it into a matrix with the point pairs acting as column vectors. 
Once I have transformed the matrix, I can extract the new list of
points using the columns() method.
The other item I improvised, since I wanted to "show" a list of
polygons generated in a loop is based on your  show(p1 + p2 + p3). 
I create a running "sum" of polygons using 
     polysum+=polygon(p)
That actually works.  I don't know how to visualize the structure
of polysum except to see it as a sum or concatenation of a series of
polygons.
--David Chandler

On Tue, Apr 2, 2013 at 7:51 PM, michel paul wrote:
See if this [12] helps. I was investigating these kinds of things

awhile back.
I've only used a rotation matrix here, but it would of course be
possible to add other transformations in this kind of scheme.
To plot pre-image and image together, add them in a show() function.
- Michel

On Tue, Apr 2, 2013 at 12:14 PM, David Chandler wrote:
I think there is some Sage expertise in this forum.  I need some
Sage advice.
I am trying to do a unit on geometric transformations using
matrices.  The polygon vertices are the column vectors of a matrix. 
I want to hit the polygon repeatedly with a transformation matrix (for
instance one that does a rotation combined with a dilation), then I
want to plot the original polygon and the sequence of transformed
polygons all on the same plot.  (The particular case I am describing
should produce a spiral pattern.)
I can see how to create matrices and transform them in Sage, and I can
see how to create polygons, and plot them, but I don't see how to
create matrices that represent the polygon or transform a polygon with
a matrix or convert a polygon to a vertex matrix or convert a matrix
to a polygon.  Plotting the pre-image and the image polygons all on
the same graph is another issue.
--David Chandler
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--
===================================
"What I cannot create, I do not understand."
- Richard Feynman===================================
"Computer science is the new mathematics."
- Dr. Christos Papadimitriou
===================================
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--
===================================
"What I cannot create, I do not understand."
- Richard Feynman===================================
"Computer science is the new mathematics."
- Dr. Christos Papadimitriou
===================================
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Links:
------
[1]
http://aleph.sagemath.org/?z=eJydVcFuozAQvSPxD6P2UEgpDamq7VbisNpe9rCXNrcoqh
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lang=sage[3]
http://aleph.sagemath.org/?z=eJxtVE1v2zAMvQfIfyDaQ-zOCdoMPayDT-2OBYYutyAYFF
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mp;lang=sage[5] http://aleph.sagemath.org
[6] http://sagenb.org
[7]
http://aleph.sagemath.org/?z=eJxdj8GKAjEMhu-C75CbqRZ39CRCwfcYBul2qhZnJiGt2M
c3unNYzOVPIHxfcoI0lSg-lOWijxdgGryguGpz8VJco0ns9mtOtq-O08_u0JjjcgFaXDI4aFGwm
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m1nBb4Ue05mZeKMnyv-BSd_MqDszdPU7-HBfmY3aLfjMMZSz-JLI7cwLrcVWkQ==&lang=s
age[12]
http://aleph.sagemath.org/?z=eJytksFqxCAQhu-BvIPQy9jakC30siD0GXoNIUjiboRERW
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iTcRxiBogLcXOgTEx2FLyuXulTtpcrwv0NwYS3ssfOCVSGH0JRXyPv-VE=&lang=sage[16] http://groups.google.com/group/mathfuture?hl=en
[17] https://groups.google.com/groups/opt_out
[20] http://groups.google.com/group/mathfuture?hl=en
[21] https://groups.google.com/groups/opt_out
[24] http://groups.google.com/group/mathfuture?hl=en
[25] https://groups.google.com/groups/opt_out
[28] http://groups.google.com/group/mathfuture?hl=en
[29] https://groups.google.com/groups/opt_out
[30] http://groups.google.com/group/mathfuture?hl=en
[31] https://groups.google.com/groups/opt_out

[michel paul]

I don't know how long the temporary link lasts. I haven't checked that yet.

I was working with the original one you sent and re-organized it to be interactive. It is here.

This is turning into a really cool thing! 

- Michel

[David Chandler]

That certainly is concise!  I wasn't aware that Sage would exponentiate a matrix.

--David Chandler

[michel paul]

On Wed, Apr 3, 2013 at 11:11 PM, David Chandler <david...@gmail.com> wrote:

That certainly is concise!  I wasn't aware that Sage would exponentiate a matrix.

--David Chandler

Yeah, Sage tries to be as mathematical as possible. If m is a square matrix, m^(-1) is its multiplicative inverse. You can also find it in an object oriented way as m.inverse().

I must say, this is a wonderful way to illustrate a geometric sequence! Here is the latest iteration. I moved the number of vertices out of the parameter list. That way the user can change rotation and scale and number of copies while keeping the original vertices. Otherwise, everything gets changed if all you want is to tweak something or study what happens with a different rotation.

I also made the axes on/off a toggle. Sometimes it's nice just to see the pattern.

I haven't yet worked on making your last file interactive, but it would be very similar, just incorporating a few more details. 

[michel paul]

This has become serious fun. With this single cell server, all kinds of possibilities open up for communicating this stuff.

And that's the REAL reason programming belongs in education - it's a new way of communicating ideas between humans, not just from humans to machines.

One problem I can relate from experience - trying to use this server in a lab with a bunch of students at once can produce a lag. That's also true of the Sage public servers.

In creating random polygons I thought it would be good to have the initial polygon situated with one base on the x-axis, so I create two vertices with y=0. I thought that might help students understand the rotation. However, that's very easily modified. That's the beauty of this framework. An idea can evolve in all kinds of ways with simple user modifications. Yeah, this is the new Algebra! 

I also changed 'theta' to 'radians' to name the variable for the angle of rotation. I thought that would be good for an interactive prompt. Sage thinks in terms of radians.

Some questions for students to consider as they explore using this -

• What kind of result would you expect from setting radians = 0? Try to visualize/understand it first! Then see what happens.
• Can you explain what's happening when radians = 0?

• What kind of result would you expect from setting radians = pi? You know the drill ... 
  
• From setting radians = pi/2?
  
• radians = pi/4?
  
• What kind of result would you expect from setting scale = 1? 
  
• What kind of result would you expect from setting scale > 1? 
• Why should you be careful with the number of copies before you run it with scale > 1? : )

Any others?

Yeah, your fully implemented transformational tool kit can definitely be turned interactive. It pretty much just requires using the '@interact' decorator prior to a function definition. Any parameters you initialize will become input boxes. Initializing with different kinds of tuples or arrays will produce sliders or pull down menus.  

This is cool. I'm really glad you raised this.

- Michel