7 ms·

Hexaflexagons [video] (2012)

580 points by mgdlbp 4y ago

deleted 4y ago
[deleted]
mgdlbp 4y ago
A recent submission[1] reminds me of the series on flexagons—folded paper with more than two faces that can be swapped via a 'flexing' operation—by Vi Hart, whose videos on mathematics were quite popular on YouTube around ten years ago. https://www.youtube.com/user/Vihart/videos?sort=p https://www.youtube.com/user/Vihart/videos?sort=p, https://hn.algolia.com/?query=vi%20hart https://hn.algolia.com/?query=vi%20hart [1] 'folding' in a different sense: How to fold a Julia fractal (2013) - https://news.ycombinator.com/item?id=32209192 https://news.ycombinator.com/item?id=32209192
- mgdlbp 4y ago
  Interesting— Vi Hart was a PI for YC Research's HARC (known mostly for Bret Victor's Dynamicland) – small world https://hn.algolia.com/?query=harc https://hn.algolia.com/?query=harc
  - O__________O 4y ago
    Yep, Vi Hart even posted single comment on HN years ago: https://news.ycombinator.com/threads?id=vihart https://news.ycombinator.com/threads?id=vihart
notRobot 4y ago
I know about Vi Hart but hadn't seen this series before, I just watched the first two videos and my mind is so blown right now, this is incredible. I love fun math stuff like this.
- Jtsummers 4y ago
  If you liked this, you ought to check out Gardner's articles themselves: https://en.wikipedia.org/wiki/Martin_Gardner_bibliography#%22Mathematical_Games%22:_The_Scientific_American_columns https://en.wikipedia.org/wiki/Martin_Gardner_bibliography#%2... - That's a list of the books (and on that page many other things) collecting Gardner's Scientific American "Mathematical Games" articles. https://www.cambridge.org/us/academic/subjects/mathematics/series/new-martin-gardner-mathematical-library https://www.cambridge.org/us/academic/subjects/mathematics/s... - First 4 of the reprints available here (or a bookseller of your choice).
Klaster_1 4y ago
Are there any practical applications to this, like making nanoscale structures that can change particle or surface properties? What's the smallest possible structure that enables bending? This looks like one of those mundane problems that yields unexpectedly impactful applications, like moiré pattern and magic angle graphene.
- tocs3 4y ago
  I do not how practical but maybe a business card with different bits of information depending on how it is folded.
- StephenAmar 4y ago
  I can think of https://youtu.be/ThwuT3_AG6w https://youtu.be/ThwuT3_AG6w
nightchalk16 4y ago
http://loki3.com/flex/ http://loki3.com/flex/
mdp2021 4y ago
I built so many, as a child, after reading Martin Gardner. Six faces for the basic ones, then many more. I used the tape of calculators with mechanical printing. Edit: looking for the original article from Martin Gardner; meanwhile: the submitted video was commented by the Scientific American (the original container) - https://blogs.scientificamerican.com/observations/flexagon-but-not-forgotten/ https://blogs.scientificamerican.com/observations/flexagon-b... The original article was Flexagons // In which strips of paper are used to make hexagonal figures with unusual properties, published on December 1, 1956. The Scientific American has it online, but paywalled. Found it: https://archive.org/details/martingardnerthecolossalbookofmathematics/page/n401/mode/2up https://archive.org/details/martingardnerthecolossalbookofma... in Martin Gardner - The Colossal Book Of Mathematics (page 385)
washyourdishes 4y ago
I love to recommend ViHart’s “Suspend your Disbelief” video: https://youtu.be/deg1wmYjwtk https://youtu.be/deg1wmYjwtk It takes a special place in my heart for being able to break apart and explain the effects of media and belief - especially of media referenced within media.
- sn9 4y ago
  This is one of those rare videos that makes me feel less alone by describing something it seems like everyone online is unaware of.
- JKCalhoun 4y ago
  I had not seen that one before. She truly is genius.
- Nebasuke 4y ago
  Thanks for linking this, I loved this video. I'm surprised it's relatively unpopular compared to the rest of the channel, but I suppose that supports her point...
- Karellen 4y ago
  Also, "How I Feel About Logarithms": https://youtu.be/N-7tcTIrers https://youtu.be/N-7tcTIrers Genius.
  - qwertox 4y ago
    "Sometimes, to make the harder things simple, you first need to make the simple things harder"
O__________O 4y ago
(2012) is date for videos from this thread. Happy to see Vi Hart’s still making videos: https://m.youtube.com/user/Vihart/videos https://m.youtube.com/user/Vihart/videos
- deleted 4y ago
  [deleted]
sam_goody 4y ago
I saw a reference to Martin Gardner's article when I was in, maybe, 9th grade. So I went to the main branch of the NY public library ("reading between the lions") and looked up the article on microfiche. It was fascinating and fun - from how to make a row of equilateral triangles with nothing more than a ruler, pen and paper. Many, many hours were spent on the various flexagon derivatives. Years later, I made a JS page to combine images and print hexaflexagons for my son's classmates. Had some fun with that as well; it was much more confounding than I expected. Nowadays, microfiche is gone, the public library is unfortunately not in most kids vernacular, and the attention span that would be required to enjoy doing the math I did has been hijacked. (TikTok !== figure-out-flexagon)
- marcelluspye 4y ago
  [flagged]
  - collegeburner 4y ago
    your comment is agephobic
- O__________O 4y ago
  TikTok has a lot of creative content on flexagons: https://www.tiktok.com/tag/flexagon https://www.tiktok.com/tag/flexagon It’s also not intended for long-form content, which is fine. If someone is super interested in the topic, related math, etc — easy to find information elsewhere using search engine and the short videos are much more likely to get someone’s attention.
  - sam_goody 4y ago
    Oh sure, the internet has a lot more related info readily available than was ever possible by using microfiche. And my phone can keep track of more more phone numbers than I was ever able to memorize. But, I know far fewer numbers now than I did as a kid, and my son will know far less than I did about flexagons. One reason may be because he knows it is readily available, he won't learn it himself. And one reason is because there is so much surface level learning available, which is so easy and interesting to consume, that anything that takes work and time is postponed, and eventually conditioned to be ignored. This is true for porn, fast-food, caffeine, or what have you - easy, gratifying, effortless dopamine rushes will destroy the positive (and ultimately more gratifying) activities that the brain chemicals were supposed to encourage in the first place. As someone who works with youth, the level of constant distraction that the smartphone has created is not plausibly denied.
donclark 4y ago
related: https://www.alibris.com/search/books/isbn/9781944686109?invid=17201775182 https://www.alibris.com/search/books/isbn/9781944686109?invi... https://www.etsy.com/listing/1231396190/tri-hexa-flexa-coder?gpla=1&gao=1& https://www.etsy.com/listing/1231396190/tri-hexa-flexa-coder...
deleted 4y ago
[deleted]
pixl97 4y ago
Hexagons are the bestagons https://youtu.be/thOifuHs6eY https://youtu.be/thOifuHs6eY
- O__________O 4y ago
  To be fair, hexagons alone are unable to form a sphere like surface: https://m.youtube.com/watch?v=btPqKAGyajM https://m.youtube.com/watch?v=btPqKAGyajM And of course the odd response from the UK government to the above petition was to basically say having a realistic football signs would increase the odds of accidents, lol: https://aperiodical.com/2017/10/standupmaths-petition-has-had-a-response-from-the-government/ https://aperiodical.com/2017/10/standupmaths-petition-has-ha...
  - Aardwolf 4y ago
    "The purpose of a traffic sign is not to raise public appreciation and awareness of geometry which is better dealt with in other ways." Maybe the petition was done the wrong way: It's not about geometry lessens. Soccer balls literally don't look like that. There are (most likely) way more people appreciating sports than geometry in the UK, so that'd be a better argument
    - O__________O 4y ago
      Agree. Video mentions that sports fans likely care and that they only wanted future signed updated, the petition though as far as I am able to tell mentions neither of these points: https://petition.parliament.uk/archived/petitions/202305 https://petition.parliament.uk/archived/petitions/202305
Despoisj 4y ago
I made a 6 sided square flexagon puzzle for my dad, it's surprisingly simple yet can take forever to solve. If interested, I've spend quite some time looking for more square flexagons, and made one with 12 and another with 14 faces (along with a puzzle version)! Tricky to build properly but stunning.
CliffStoll 4y ago
Vi Hart's videos -- terrific!
tunesmith 4y ago
The sixth video in the series is really a work of art: https://www.youtube.com/watch?v=DIyruYQ-N4Q&list=PLaNzoFtkQ7rbt5ac9qdi76iNKuqZWQkB3&index=6 https://www.youtube.com/watch?v=DIyruYQ-N4Q&list=PLaNzoFtkQ7... I've thought a lot about arguments and premises and values and conclusions and how it connects with frustration and humiliation and shame, and really wasn't expecting all those subjects to come in in a video about hexaflexagons.
- someweirdperson 4y ago
  Quote from part 6: "The pink side with the yellow center is a theoretical possibility that remains unobserved as in practice it seems to be unopenupable."
  - lupire 4y ago
    Strangely, in the Gardner article that she used for the outline of her video, it explicitly states that its eqivalent of the pink side with yellow center is theoretically impossible, in the hexahexafrustragon.
bmorton 4y ago
ge96 4y ago
Damn... I have not seen this in a long time
stevenjgarner 4y ago
She had me at you "find that your new American paper doesn't fit in your old English binder."
2muchcoffeeman 4y ago
I thought Feynman Diagrams were for interactions of subatomic particles.
omneity 4y ago
Maybe someone with more specialized knowledge can shime in, but this looks awfully similar to me to a discretized SU(1) symmetry (maybe SU(2)? My intuition in Lie algebras is extremely rusty) If so, hexaflexagons (or general flexagons) could present interesting analogies to Calabi-Yau manifolds, especially in terms of holonomic properties, which might yield some insights or parallels for type II string theory. Or in more intuitive terms as I see it, a flexagon looks suspiciously similar to a string oscillating on a folded dimension, but discretized as some sort of torus/state machine. (not a physicist, worked at a HEP lab ages ago, I hope some of my comment made sense)
anandology 4y ago
When I saw the name Vi Hart, the author of the video, it sounded quite familiar. It turned out that She is the daughter of George W Hart, a mathematician and polyhedron man. I'm a big fan of work zome geometry and amazing polyhedron. If you are interested in recreational mathematics, don't miss to checkout https://georgehart.com/ https://georgehart.com/.
AndrianV 4y ago
I don't know why I had this image in my head, but I pictured Stone, Tuckerman, and Feynmann as being around the same age when this took place. Imagine how disheartened I was to learn that they were 23 years old, 24 years old, and 21 years old correspondingly.

580 points by mgdlbp 4y ago

deleted 4y ago

[deleted]

A recent submission[1] reminds me of the series on flexagons—folded paper with more than two faces that can be swapped via a 'flexing' operation—by Vi Hart, whose videos on mathematics were quite popular on YouTube around ten years ago. https://www.youtube.com/user/Vihart/videos?sort=p https://www.youtube.com/user/Vihart/videos?sort=p, https://hn.algolia.com/?query=vi%20hart https://hn.algolia.com/?query=vi%20hart [1] 'folding' in a different sense: How to fold a Julia fractal (2013) - https://news.ycombinator.com/item?id=32209192 https://news.ycombinator.com/item?id=32209192

mgdlbp 4y ago
Interesting— Vi Hart was a PI for YC Research's HARC (known mostly for Bret Victor's Dynamicland) – small world https://hn.algolia.com/?query=harc https://hn.algolia.com/?query=harc
- O__________O 4y ago
  Yep, Vi Hart even posted single comment on HN years ago: https://news.ycombinator.com/threads?id=vihart https://news.ycombinator.com/threads?id=vihart

notRobot 4y ago

I know about Vi Hart but hadn't seen this series before, I just watched the first two videos and my mind is so blown right now, this is incredible. I love fun math stuff like this.

Jtsummers 4y ago
If you liked this, you ought to check out Gardner's articles themselves: https://en.wikipedia.org/wiki/Martin_Gardner_bibliography#%22Mathematical_Games%22:_The_Scientific_American_columns https://en.wikipedia.org/wiki/Martin_Gardner_bibliography#%2... - That's a list of the books (and on that page many other things) collecting Gardner's Scientific American "Mathematical Games" articles. https://www.cambridge.org/us/academic/subjects/mathematics/series/new-martin-gardner-mathematical-library https://www.cambridge.org/us/academic/subjects/mathematics/s... - First 4 of the reprints available here (or a bookseller of your choice).

Klaster_1 4y ago

Are there any practical applications to this, like making nanoscale structures that can change particle or surface properties? What's the smallest possible structure that enables bending? This looks like one of those mundane problems that yields unexpectedly impactful applications, like moiré pattern and magic angle graphene.

tocs3 4y ago
I do not how practical but maybe a business card with different bits of information depending on how it is folded.
StephenAmar 4y ago
I can think of https://youtu.be/ThwuT3_AG6w https://youtu.be/ThwuT3_AG6w

nightchalk16 4y ago

http://loki3.com/flex/ http://loki3.com/flex/

mdp2021 4y ago

I built so many, as a child, after reading Martin Gardner. Six faces for the basic ones, then many more. I used the tape of calculators with mechanical printing. Edit: looking for the original article from Martin Gardner; meanwhile: the submitted video was commented by the Scientific American (the original container) - https://blogs.scientificamerican.com/observations/flexagon-but-not-forgotten/ https://blogs.scientificamerican.com/observations/flexagon-b... The original article was Flexagons // In which strips of paper are used to make hexagonal figures with unusual properties, published on December 1, 1956. The Scientific American has it online, but paywalled. Found it: https://archive.org/details/martingardnerthecolossalbookofmathematics/page/n401/mode/2up https://archive.org/details/martingardnerthecolossalbookofma... in Martin Gardner - The Colossal Book Of Mathematics (page 385)

washyourdishes 4y ago

I love to recommend ViHart’s “Suspend your Disbelief” video: https://youtu.be/deg1wmYjwtk https://youtu.be/deg1wmYjwtk It takes a special place in my heart for being able to break apart and explain the effects of media and belief - especially of media referenced within media.

sn9 4y ago
This is one of those rare videos that makes me feel less alone by describing something it seems like everyone online is unaware of.
JKCalhoun 4y ago
I had not seen that one before. She truly is genius.
Nebasuke 4y ago
Thanks for linking this, I loved this video. I'm surprised it's relatively unpopular compared to the rest of the channel, but I suppose that supports her point...
Karellen 4y ago
Also, "How I Feel About Logarithms": https://youtu.be/N-7tcTIrers https://youtu.be/N-7tcTIrers Genius.
- qwertox 4y ago
  "Sometimes, to make the harder things simple, you first need to make the simple things harder"

O__________O 4y ago

(2012) is date for videos from this thread. Happy to see Vi Hart’s still making videos: https://m.youtube.com/user/Vihart/videos https://m.youtube.com/user/Vihart/videos

deleted 4y ago
[deleted]

sam_goody 4y ago

I saw a reference to Martin Gardner's article when I was in, maybe, 9th grade. So I went to the main branch of the NY public library ("reading between the lions") and looked up the article on microfiche. It was fascinating and fun - from how to make a row of equilateral triangles with nothing more than a ruler, pen and paper. Many, many hours were spent on the various flexagon derivatives. Years later, I made a JS page to combine images and print hexaflexagons for my son's classmates. Had some fun with that as well; it was much more confounding than I expected. Nowadays, microfiche is gone, the public library is unfortunately not in most kids vernacular, and the attention span that would be required to enjoy doing the math I did has been hijacked. (TikTok !== figure-out-flexagon)

marcelluspye 4y ago
[flagged]
- collegeburner 4y ago
  your comment is agephobic
O__________O 4y ago
TikTok has a lot of creative content on flexagons: https://www.tiktok.com/tag/flexagon https://www.tiktok.com/tag/flexagon It’s also not intended for long-form content, which is fine. If someone is super interested in the topic, related math, etc — easy to find information elsewhere using search engine and the short videos are much more likely to get someone’s attention.
- sam_goody 4y ago
  Oh sure, the internet has a lot more related info readily available than was ever possible by using microfiche. And my phone can keep track of more more phone numbers than I was ever able to memorize. But, I know far fewer numbers now than I did as a kid, and my son will know far less than I did about flexagons. One reason may be because he knows it is readily available, he won't learn it himself. And one reason is because there is so much surface level learning available, which is so easy and interesting to consume, that anything that takes work and time is postponed, and eventually conditioned to be ignored. This is true for porn, fast-food, caffeine, or what have you - easy, gratifying, effortless dopamine rushes will destroy the positive (and ultimately more gratifying) activities that the brain chemicals were supposed to encourage in the first place. As someone who works with youth, the level of constant distraction that the smartphone has created is not plausibly denied.

donclark 4y ago

related: https://www.alibris.com/search/books/isbn/9781944686109?invid=17201775182 https://www.alibris.com/search/books/isbn/9781944686109?invi... https://www.etsy.com/listing/1231396190/tri-hexa-flexa-coder?gpla=1&gao=1& https://www.etsy.com/listing/1231396190/tri-hexa-flexa-coder...

deleted 4y ago

[deleted]

pixl97 4y ago

Hexagons are the bestagons https://youtu.be/thOifuHs6eY https://youtu.be/thOifuHs6eY

O__________O 4y ago
To be fair, hexagons alone are unable to form a sphere like surface: https://m.youtube.com/watch?v=btPqKAGyajM https://m.youtube.com/watch?v=btPqKAGyajM And of course the odd response from the UK government to the above petition was to basically say having a realistic football signs would increase the odds of accidents, lol: https://aperiodical.com/2017/10/standupmaths-petition-has-had-a-response-from-the-government/ https://aperiodical.com/2017/10/standupmaths-petition-has-ha...
- Aardwolf 4y ago
  "The purpose of a traffic sign is not to raise public appreciation and awareness of geometry which is better dealt with in other ways." Maybe the petition was done the wrong way: It's not about geometry lessens. Soccer balls literally don't look like that. There are (most likely) way more people appreciating sports than geometry in the UK, so that'd be a better argument
  - O__________O 4y ago
    Agree. Video mentions that sports fans likely care and that they only wanted future signed updated, the petition though as far as I am able to tell mentions neither of these points: https://petition.parliament.uk/archived/petitions/202305 https://petition.parliament.uk/archived/petitions/202305

Despoisj 4y ago

I made a 6 sided square flexagon puzzle for my dad, it's surprisingly simple yet can take forever to solve. If interested, I've spend quite some time looking for more square flexagons, and made one with 12 and another with 14 faces (along with a puzzle version)! Tricky to build properly but stunning.

CliffStoll 4y ago

Vi Hart's videos -- terrific!

tunesmith 4y ago

The sixth video in the series is really a work of art: https://www.youtube.com/watch?v=DIyruYQ-N4Q&list=PLaNzoFtkQ7rbt5ac9qdi76iNKuqZWQkB3&index=6 https://www.youtube.com/watch?v=DIyruYQ-N4Q&list=PLaNzoFtkQ7... I've thought a lot about arguments and premises and values and conclusions and how it connects with frustration and humiliation and shame, and really wasn't expecting all those subjects to come in in a video about hexaflexagons.

someweirdperson 4y ago
Quote from part 6: "The pink side with the yellow center is a theoretical possibility that remains unobserved as in practice it seems to be unopenupable."
- lupire 4y ago
  Strangely, in the Gardner article that she used for the outline of her video, it explicitly states that its eqivalent of the pink side with yellow center is theoretically impossible, in the hexahexafrustragon.

bmorton 4y ago

ge96 4y ago

Damn... I have not seen this in a long time

stevenjgarner 4y ago

She had me at you "find that your new American paper doesn't fit in your old English binder."

2muchcoffeeman 4y ago

I thought Feynman Diagrams were for interactions of subatomic particles.

omneity 4y ago

Maybe someone with more specialized knowledge can shime in, but this looks awfully similar to me to a discretized SU(1) symmetry (maybe SU(2)? My intuition in Lie algebras is extremely rusty) If so, hexaflexagons (or general flexagons) could present interesting analogies to Calabi-Yau manifolds, especially in terms of holonomic properties, which might yield some insights or parallels for type II string theory. Or in more intuitive terms as I see it, a flexagon looks suspiciously similar to a string oscillating on a folded dimension, but discretized as some sort of torus/state machine. (not a physicist, worked at a HEP lab ages ago, I hope some of my comment made sense)

anandology 4y ago

When I saw the name Vi Hart, the author of the video, it sounded quite familiar. It turned out that She is the daughter of George W Hart, a mathematician and polyhedron man. I'm a big fan of work zome geometry and amazing polyhedron. If you are interested in recreational mathematics, don't miss to checkout https://georgehart.com/ https://georgehart.com/.

AndrianV 4y ago

I don't know why I had this image in my head, but I pictured Stone, Tuckerman, and Feynmann as being around the same age when this took place. Imagine how disheartened I was to learn that they were 23 years old, 24 years old, and 21 years old correspondingly.