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mathematicians computer graphic-ians — a little? help please

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Prof David West

mathematicians computer graphic-ians — a little? help please

First,

Just finished reading, the crest of the peacock (ibid lowercase), by George Gheverghese Joseph. Subtitle is "non-European roots of mathematics." Wonderful book, highest recommendation and not just to mathematicians.

My three biggest shames in life: losing my fluency in Japanese and Arabic; and excepting one course in knot theory at UW-Madison, stopping my math education at calculus in high school. I still love reading about math and mathematicians but wish I understood more.

To the question/help request. Some roots of my problem:

One) I am studying origami and specifically the way you can, in 2-dimensions, draw the pattern of folds that will yield a specific 3-D figure. And there are 'families' of 2-D patterns that an origami expert can look at and tell you if the eventual 3-D figure will have 2, 3, or 4 legs. How it is possible to 'see', in your mind, the 3-D in the 2-D?

Two) a quick look at several animated hyper-cubes show the 'interior' cube remaining cubical as the hypercube is manipulated. Must this always be true, must the six facets of the 3-D cube remain perfect squares? What degrees of freedom are allowed the various vertices of the hyper-cube?

Three) can find static hyper— for the five platonic solids, but not animations. Is it possible to provide something analogous to the hypercube animation for the other solids? I think this is a problem in manifolds as many of you have talked about.

Question: If one had a series of very vivid, very convincing, visions of animated hyper-platonic solids with almost complete freedom of movement of the various vertices (doesn't really apply to hypersphere) — how would one go about finding visualizations that would assist in confirming/denying/making sense of the visions?

Please forgive the crude way of expressing/asking my question. I am both math and computer graphic ignorant.

davew

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Frank Wimberly-2

Re: mathematicians computer graphic-ians — a little? help please

Is this the hypercube video you saw?

https://youtu.be/RqQvVts5Yj0

---
Frank C. Wimberly
140 Calle Ojo Feliz,
Santa Fe, NM 87505

505 670-9918
Santa Fe, NM

On Thu, Jun 4, 2020, 8:22 PM Prof David West <[hidden email]> wrote:

First,

Just finished reading, the crest of the peacock (ibid lowercase), by George Gheverghese Joseph. Subtitle is "non-European roots of mathematics." Wonderful book, highest recommendation and not just to mathematicians.

My three biggest shames in life: losing my fluency in Japanese and Arabic; and excepting one course in knot theory at UW-Madison, stopping my math education at calculus in high school. I still love reading about math and mathematicians but wish I understood more.

To the question/help request. Some roots of my problem:

One) I am studying origami and specifically the way you can, in 2-dimensions, draw the pattern of folds that will yield a specific 3-D figure. And there are 'families' of 2-D patterns that an origami expert can look at and tell you if the eventual 3-D figure will have 2, 3, or 4 legs. How it is possible to 'see', in your mind, the 3-D in the 2-D?

Two) a quick look at several animated hyper-cubes show the 'interior' cube remaining cubical as the hypercube is manipulated. Must this always be true, must the six facets of the 3-D cube remain perfect squares? What degrees of freedom are allowed the various vertices of the hyper-cube?

Three) can find static hyper— for the five platonic solids, but not animations. Is it possible to provide something analogous to the hypercube animation for the other solids? I think this is a problem in manifolds as many of you have talked about.

Question: If one had a series of very vivid, very convincing, visions of animated hyper-platonic solids with almost complete freedom of movement of the various vertices (doesn't really apply to hypersphere) — how would one go about finding visualizations that would assist in confirming/denying/making sense of the visions?

Please forgive the crude way of expressing/asking my question. I am both math and computer graphic ignorant.

davew
- .... . -..-. . ...- --- .-.. ..- - .. --- -. -..-. .-- .. .-.. .-.. -..-. -... . -..-. .-.. .. ...- . -..-. ... - .-. . .- -- . -..
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Prof David West

Re: mathematicians computer graphic-ians — a little? help please

yes. I did see another that seemed to show the cube in the center not retaining its squares

On Thu, Jun 4, 2020, at 8:57 PM, Frank Wimberly wrote:

Is this the hypercube video you saw?

https://youtu.be/RqQvVts5Yj0
---
Frank C. Wimberly
140 Calle Ojo Feliz,
Santa Fe, NM 87505

505 670-9918
Santa Fe, NM

On Thu, Jun 4, 2020, 8:22 PM Prof David West <[hidden email]> wrote:

First,

Just finished reading, the crest of the peacock (ibid lowercase), by George Gheverghese Joseph. Subtitle is "non-European roots of mathematics." Wonderful book, highest recommendation and not just to mathematicians.

My three biggest shames in life: losing my fluency in Japanese and Arabic; and excepting one course in knot theory at UW-Madison, stopping my math education at calculus in high school. I still love reading about math and mathematicians but wish I understood more.

To the question/help request. Some roots of my problem:

One) I am studying origami and specifically the way you can, in 2-dimensions, draw the pattern of folds that will yield a specific 3-D figure. And there are 'families' of 2-D patterns that an origami expert can look at and tell you if the eventual 3-D figure will have 2, 3, or 4 legs. How it is possible to 'see', in your mind, the 3-D in the 2-D?

Two) a quick look at several animated hyper-cubes show the 'interior' cube remaining cubical as the hypercube is manipulated. Must this always be true, must the six facets of the 3-D cube remain perfect squares? What degrees of freedom are allowed the various vertices of the hyper-cube?

Three) can find static hyper— for the five platonic solids, but not animations. Is it possible to provide something analogous to the hypercube animation for the other solids? I think this is a problem in manifolds as many of you have talked about.

Question: If one had a series of very vivid, very convincing, visions of animated hyper-platonic solids with almost complete freedom of movement of the various vertices (doesn't really apply to hypersphere) — how would one go about finding visualizations that would assist in confirming/denying/making sense of the visions?

Please forgive the crude way of expressing/asking my question. I am both math and computer graphic ignorant.

davew
- .... . -..-. . ...- --- .-.. ..- - .. --- -. -..-. .-- .. .-.. .-.. -..-. -... . -..-. .-.. .. ...- . -..-. ... - .-. . .- -- . -..
FRIAM Applied Complexity Group listserv
Zoom Fridays 9:30a-12p Mtn GMT-6 bit.ly/virtualfriam
un/subscribe http://redfish.com/mailman/listinfo/friam_redfish.com
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- .... . -..-. . ...- --- .-.. ..- - .. --- -. -..-. .-- .. .-.. .-.. -..-. -... . -..-. .-.. .. ...- . -..-. ... - .-. . .- -- . -..
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Frank Wimberly-2

Re: mathematicians computer graphic-ians — a little? help please

As soon as the sides aren't squares it isn't a cube. Maybe that's why they use the word tesseract.

Frank

---
Frank C. Wimberly
140 Calle Ojo Feliz,
Santa Fe, NM 87505

505 670-9918
Santa Fe, NM

On Thu, Jun 4, 2020, 9:15 PM Prof David West <[hidden email]> wrote:

yes. I did see another that seemed to show the cube in the center not retaining its squares

On Thu, Jun 4, 2020, at 8:57 PM, Frank Wimberly wrote:

Is this the hypercube video you saw?

https://youtu.be/RqQvVts5Yj0
---
Frank C. Wimberly
140 Calle Ojo Feliz,
Santa Fe, NM 87505

505 670-9918
Santa Fe, NM

On Thu, Jun 4, 2020, 8:22 PM Prof David West <[hidden email]> wrote:

First,

Just finished reading, the crest of the peacock (ibid lowercase), by George Gheverghese Joseph. Subtitle is "non-European roots of mathematics." Wonderful book, highest recommendation and not just to mathematicians.

My three biggest shames in life: losing my fluency in Japanese and Arabic; and excepting one course in knot theory at UW-Madison, stopping my math education at calculus in high school. I still love reading about math and mathematicians but wish I understood more.

To the question/help request. Some roots of my problem:

One) I am studying origami and specifically the way you can, in 2-dimensions, draw the pattern of folds that will yield a specific 3-D figure. And there are 'families' of 2-D patterns that an origami expert can look at and tell you if the eventual 3-D figure will have 2, 3, or 4 legs. How it is possible to 'see', in your mind, the 3-D in the 2-D?

Two) a quick look at several animated hyper-cubes show the 'interior' cube remaining cubical as the hypercube is manipulated. Must this always be true, must the six facets of the 3-D cube remain perfect squares? What degrees of freedom are allowed the various vertices of the hyper-cube?

Three) can find static hyper— for the five platonic solids, but not animations. Is it possible to provide something analogous to the hypercube animation for the other solids? I think this is a problem in manifolds as many of you have talked about.

Question: If one had a series of very vivid, very convincing, visions of animated hyper-platonic solids with almost complete freedom of movement of the various vertices (doesn't really apply to hypersphere) — how would one go about finding visualizations that would assist in confirming/denying/making sense of the visions?

Please forgive the crude way of expressing/asking my question. I am both math and computer graphic ignorant.

davew
- .... . -..-. . ...- --- .-.. ..- - .. --- -. -..-. .-- .. .-.. .-.. -..-. -... . -..-. .-.. .. ...- . -..-. ... - .-. . .- -- . -..
FRIAM Applied Complexity Group listserv
Zoom Fridays 9:30a-12p Mtn GMT-6 bit.ly/virtualfriam
un/subscribe http://redfish.com/mailman/listinfo/friam_redfish.com
archives: http://friam.471366.n2.nabble.com/
FRIAM-COMIC http://friam-comic.blogspot.com/
- .... . -..-. . ...- --- .-.. ..- - .. --- -. -..-. .-- .. .-.. .-.. -..-. -... . -..-. .-.. .. ...- . -..-. ... - .-. . .- -- . -..
FRIAM Applied Complexity Group listserv
Zoom Fridays 9:30a-12p Mtn GMT-6 bit.ly/virtualfriam
un/subscribe http://redfish.com/mailman/listinfo/friam_redfish.com
archives: http://friam.471366.n2.nabble.com/
FRIAM-COMIC http://friam-comic.blogspot.com/

- .... . -..-. . ...- --- .-.. ..- - .. --- -. -..-. .-- .. .-.. .-.. -..-. -... . -..-. .-.. .. ...- . -..-. ... - .-. . .- -- . -..
FRIAM Applied Complexity Group listserv
Zoom Fridays 9:30a-12p Mtn GMT-6 bit.ly/virtualfriam
un/subscribe http://redfish.com/mailman/listinfo/friam_redfish.com
archives: http://friam.471366.n2.nabble.com/
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jon zingale

Re: mathematicians computer graphic-ians — a little? help please

In reply to this post by Prof David West

Dave,

Sorry-not-sorry to bring up Penrose again. One of my favorite

mathematical facts is that generalized Penrose tilings can be seen

as a consequence of projecting a 5-dimensional lattice into the

plane, where this lattice can be imagined as a tiling of hyper-cubes.

I am not sure this is helpful, but it is pretty cool. More to your goals:

two) I suspect not, but tell me more. Can you constrain the question?

three) Perhaps showing my age, less than 8 years ago I had to prove

for Dr. Starbird's class the theorem that there are only 5 regular

Platonic solids. What was interesting about the proof is that no

more than graph theory and algebra is needed. To some extent,

it is what is beautiful about these really early theorems.

Jon

ps. I prefer Gmail for editing together a post.

Steve Smith

Re: mathematicians computer graphic-ians — a little? help please

In reply to this post by Frank Wimberly-2

First,

Just finished reading, the crest of the peacock (ibid lowercase), by George Gheverghese Joseph. Subtitle is "non-European roots of mathematics." Wonderful book, highest recommendation and not just to mathematicians.

My three biggest shames in life: losing my fluency in Japanese and Arabic; and excepting one course in knot theory at UW-Madison, stopping my math education at calculus in high school. I still love reading about math and mathematicians but wish I understood more.

To the question/help request. Some roots of my problem:

One) I am studying origami and specifically the way you can, in 2-dimensions, draw the pattern of folds that will yield a specific 3-D figure. And there are 'families' of 2-D patterns that an origami expert can look at and tell you if the eventual 3-D figure will have 2, 3, or 4 legs. How it is possible to 'see', in your mind, the 3-D in the 2-D?

I've only dabbled with origami and share your implied questions about the way people who work with it a lot seem to be not only able to "guess" what a 2d pattern of folds will be in 3d but can "design" in 2d to yield 3d shapes. I suspect a formalization of how they do it is closer to group theory than geometry. As for "how is it possible?" I think that is the fundamental question for all forms of "fusing" sensory data of one type into higher level abstractions. The only way I know to acquire such a skill is to practice, practice, practice.

For highD data, that means (for me) working in as high-dimensional of a perception space as possible (e.g. stereo + motion parallax with other depth cues like texture and saturation and hue. Manipulating the object "directly" with a 3D pointer (spaceball, etc.) or better "pinch gloves" or even better, haptic-gloves (looking a bit edward scissorhandy). My best experiences with all of this have been in a modestly good VR environment (my preferred being Flatland from UNM, named after EA Abbot's Victorian Romance in Many Dimensions (for the very reason you are asking about this I'd say)) on an immersive workbench (8' diagonal view surface tilted at 20+ degrees with active stereography, head and hand tracking, and pinch gloves). You literally "reach out and grab geometry and rotate/drag it around". I'd also recommend "listening" to them, but that can be a little trickier.

Staring at clouds and other phenomena which are 3D ++ (the shape of a cloud as observed is roughly an isosurface of temperature, pressure, humidity over the three spatial dimensions) as they evolve (facilitated by timelapse and best observed as they "squeeze" over mountains or "form" over bodies of water.

Two) a quick look at several animated hyper-cubes show the 'interior' cube remaining cubical as the hypercube is manipulated. Must this always be true, must the six facets of the 3-D cube remain perfect squares? What degrees of freedom are allowed the various vertices of the hyper-cube?

The conventional projections of the Tesseract into 3D are only rotated around the yz, xz, xz axes... the additional ones that include the w axis do not present as "perfect cubes". See second :40 and on in this video: https://www.youtube.com/watch?v=fjwvMO-n2dY

It might be easier to accept this if you notice that off-axis rotations of a cube when projected into 2D yield non-square faces in 2D

Three) can find static hyper— for the five platonic solids, but not animations. Is it possible to provide something analogous to the hypercube animation for the other solids? I think this is a problem in manifolds as many of you have talked about.

The mathematical objects you are talking about are called regular convex 4-polytopes, Wikipedia has a good article on the topic:

https://en.wikipedia.org/wiki/Regular_4-polytope

Question: If one had a series of very vivid, very convincing, visions of animated hyper-platonic solids with almost complete freedom of movement of the various vertices (doesn't really apply to hypersphere) — how would one go about finding visualizations that would assist in confirming/denying/making sense of the visions?

The video above tumbles you through some regular 4 polytopes... I'll give everyone else the trigger-warning <trippy man!>

This guy: https://www.youtube.com/watch?v=2s4TqVAbfz4 has added 3D printed models frozen in mid-4D tumble to give you (maybe) some added intuition.

There are a plethora of commercial HMDs out now that would facilitate a great deal more than just staring at your laptop while geometry tumbles through 3, 4, nD. These days I bet you can drop your phone into a google-cardboard device ($3 on amazon), load up a copy of Mathematica or similar and find a program to let you tumble yourself through these experiences.

I do look forward to your "trip report" and will take you to task if *I* start dreaming in hyperspace again!

- Steve

Steve Smith

Re: mathematicians computer graphic-ians — a little? help please

and why stop at 4 when you can go higher?

https://www.youtube.com/watch?v=tfGf6gHQZQc

First,

Just finished reading, the crest of the peacock (ibid lowercase), by George Gheverghese Joseph. Subtitle is "non-European roots of mathematics." Wonderful book, highest recommendation and not just to mathematicians.

My three biggest shames in life: losing my fluency in Japanese and Arabic; and excepting one course in knot theory at UW-Madison, stopping my math education at calculus in high school. I still love reading about math and mathematicians but wish I understood more.

To the question/help request. Some roots of my problem:

One) I am studying origami and specifically the way you can, in 2-dimensions, draw the pattern of folds that will yield a specific 3-D figure. And there are 'families' of 2-D patterns that an origami expert can look at and tell you if the eventual 3-D figure will have 2, 3, or 4 legs. How it is possible to 'see', in your mind, the 3-D in the 2-D?

I've only dabbled with origami and share your implied questions about the way people who work with it a lot seem to be not only able to "guess" what a 2d pattern of folds will be in 3d but can "design" in 2d to yield 3d shapes.   I suspect a formalization of how they do it is closer to group theory than geometry.    As for "how is it possible?"   I think that is the fundamental question for all forms of "fusing" sensory data of one type into higher level abstractions. The only way I know to acquire such a skill is to practice, practice, practice.

For highD data, that means (for me) working in as high-dimensional of a perception space as possible (e.g. stereo + motion parallax with other depth cues like texture and saturation and hue.   Manipulating the object "directly" with a 3D pointer (spaceball, etc.) or better "pinch gloves" or even better, haptic-gloves (looking a bit edward scissorhandy).    My best experiences with all of this have been in a modestly good VR environment (my preferred being Flatland from UNM, named after EA Abbot's Victorian Romance in Many Dimensions (for the very reason you are asking about this I'd say)) on an immersive workbench (8' diagonal view surface tilted at 20+ degrees with active stereography, head and hand tracking, and pinch gloves). You literally "reach out and grab geometry and rotate/drag it around".   I'd also recommend "listening" to them, but that can be a little trickier.

Staring at clouds and other phenomena which are 3D ++ (the shape of a cloud as observed is roughly an isosurface of temperature, pressure, humidity over the three spatial dimensions) as they evolve (facilitated by timelapse and best observed as they "squeeze" over mountains or "form" over bodies of water.

Two) a quick look at several animated hyper-cubes show the 'interior' cube remaining cubical as the hypercube is manipulated. Must this always be true, must the six facets of the 3-D cube remain perfect squares? What degrees of freedom are allowed the various vertices of the hyper-cube?

The conventional projections of the Tesseract into 3D are only rotated around the yz, xz, xz axes... the additional ones that include the w axis do not present as "perfect cubes".   See second :40 and on in this video: https://www.youtube.com/watch?v=fjwvMO-n2dY

It might be easier to accept this if you notice that off-axis rotations of a cube when projected into 2D yield non-square faces in 2D

Three) can find static hyper— for the five platonic solids, but not animations. Is it possible to provide something analogous to the hypercube animation for the other solids? I think this is a problem in manifolds as many of you have talked about.

The mathematical objects you are talking about are called regular convex 4-polytopes, Wikipedia has a good article on the topic:

https://en.wikipedia.org/wiki/Regular_4-polytope

Question: If one had a series of very vivid, very convincing, visions of animated hyper-platonic solids with almost complete freedom of movement of the various vertices (doesn't really apply to hypersphere) — how would one go about finding visualizations that would assist in confirming/denying/making sense of the visions?

The video above tumbles you through some regular 4 polytopes... I'll give everyone else the trigger-warning <trippy man!>

This guy: https://www.youtube.com/watch?v=2s4TqVAbfz4 has added 3D printed models frozen in mid-4D tumble to give you (maybe) some added intuition.

There are a plethora of commercial HMDs out now that would facilitate a great deal more than just staring at your laptop while geometry tumbles through 3, 4, nD. These days I bet you can drop your phone into a google-cardboard device ($3 on amazon), load up a copy of Mathematica or similar and find a program to let you tumble yourself through these experiences.

I do look forward to your "trip report" and will take you to task if *I* start dreaming in hyperspace again!

- Steve
- .... . -..-. . ...- --- .-.. ..- - .. --- -. -..-. .-- .. .-.. .-.. -..-. -... . -..-. .-.. .. ...- . -..-. ... - .-. . .- -- . -..
FRIAM Applied Complexity Group listserv
Zoom Fridays 9:30a-12p Mtn GMT-6  bit.ly/virtualfriam
un/subscribe http://redfish.com/mailman/listinfo/friam_redfish.com
archives: http://friam.471366.n2.nabble.com/
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Steve Smith

Re: mathematicians computer graphic-ians — a little? help please

PPS

Given your preferences, you might want to check into Kepler's Harmonices Mundi: https://en.wikipedia.org/wiki/Harmonices_Mundi though doing it right would require becoming fluent in Latin.

and why stop at 4 when you can go higher?

https://www.youtube.com/watch?v=tfGf6gHQZQc
First,

Just finished reading, the crest of the peacock (ibid lowercase), by George Gheverghese Joseph. Subtitle is "non-European roots of mathematics." Wonderful book, highest recommendation and not just to mathematicians.

My three biggest shames in life: losing my fluency in Japanese and Arabic; and excepting one course in knot theory at UW-Madison, stopping my math education at calculus in high school. I still love reading about math and mathematicians but wish I understood more.

To the question/help request. Some roots of my problem:

One) I am studying origami and specifically the way you can, in 2-dimensions, draw the pattern of folds that will yield a specific 3-D figure. And there are 'families' of 2-D patterns that an origami expert can look at and tell you if the eventual 3-D figure will have 2, 3, or 4 legs. How it is possible to 'see', in your mind, the 3-D in the 2-D?

I've only dabbled with origami and share your implied questions about the way people who work with it a lot seem to be not only able to "guess" what a 2d pattern of folds will be in 3d but can "design" in 2d to yield 3d shapes.   I suspect a formalization of how they do it is closer to group theory than geometry.    As for "how is it possible?"   I think that is the fundamental question for all forms of "fusing" sensory data of one type into higher level abstractions. The only way I know to acquire such a skill is to practice, practice, practice.

For highD data, that means (for me) working in as high-dimensional of a perception space as possible (e.g. stereo + motion parallax with other depth cues like texture and saturation and hue.   Manipulating the object "directly" with a 3D pointer (spaceball, etc.) or better "pinch gloves" or even better, haptic-gloves (looking a bit edward scissorhandy).    My best experiences with all of this have been in a modestly good VR environment (my preferred being Flatland from UNM, named after EA Abbot's Victorian Romance in Many Dimensions (for the very reason you are asking about this I'd say)) on an immersive workbench (8' diagonal view surface tilted at 20+ degrees with active stereography, head and hand tracking, and pinch gloves). You literally "reach out and grab geometry and rotate/drag it around".   I'd also recommend "listening" to them, but that can be a little trickier.

Staring at clouds and other phenomena which are 3D ++ (the shape of a cloud as observed is roughly an isosurface of temperature, pressure, humidity over the three spatial dimensions) as they evolve (facilitated by timelapse and best observed as they "squeeze" over mountains or "form" over bodies of water.

Two) a quick look at several animated hyper-cubes show the 'interior' cube remaining cubical as the hypercube is manipulated. Must this always be true, must the six facets of the 3-D cube remain perfect squares? What degrees of freedom are allowed the various vertices of the hyper-cube?

The conventional projections of the Tesseract into 3D are only rotated around the yz, xz, xz axes... the additional ones that include the w axis do not present as "perfect cubes".   See second :40 and on in this video: https://www.youtube.com/watch?v=fjwvMO-n2dY

It might be easier to accept this if you notice that off-axis rotations of a cube when projected into 2D yield non-square faces in 2D

Three) can find static hyper— for the five platonic solids, but not animations. Is it possible to provide something analogous to the hypercube animation for the other solids? I think this is a problem in manifolds as many of you have talked about.

The mathematical objects you are talking about are called regular convex 4-polytopes, Wikipedia has a good article on the topic:

https://en.wikipedia.org/wiki/Regular_4-polytope

Question: If one had a series of very vivid, very convincing, visions of animated hyper-platonic solids with almost complete freedom of movement of the various vertices (doesn't really apply to hypersphere) — how would one go about finding visualizations that would assist in confirming/denying/making sense of the visions?

The video above tumbles you through some regular 4 polytopes... I'll give everyone else the trigger-warning <trippy man!>

This guy: https://www.youtube.com/watch?v=2s4TqVAbfz4 has added 3D printed models frozen in mid-4D tumble to give you (maybe) some added intuition.

There are a plethora of commercial HMDs out now that would facilitate a great deal more than just staring at your laptop while geometry tumbles through 3, 4, nD. These days I bet you can drop your phone into a google-cardboard device ($3 on amazon), load up a copy of Mathematica or similar and find a program to let you tumble yourself through these experiences.

I do look forward to your "trip report" and will take you to task if *I* start dreaming in hyperspace again!

- Steve
- .... . -..-. . ...- --- .-.. ..- - .. --- -. -..-. .-- .. .-.. .-.. -..-. -... . -..-. .-.. .. ...- . -..-. ... - .-. . .- -- . -..
FRIAM Applied Complexity Group listserv
Zoom Fridays 9:30a-12p Mtn GMT-6  bit.ly/virtualfriam
un/subscribe http://redfish.com/mailman/listinfo/friam_redfish.com
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- .... . -..-. . ...- --- .-.. ..- - .. --- -. -..-. .-- .. .-.. .-.. -..-. -... . -..-. .-.. .. ...- . -..-. ... - .-. . .- -- . -..
FRIAM Applied Complexity Group listserv
Zoom Fridays 9:30a-12p Mtn GMT-6  bit.ly/virtualfriam
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