Golden icosahedron
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- Опубликовано: 31 дек 2016
- This icosahedron is based on three intersecting golden rectangles. The project is easy to construct from plywood, but there are also plenty of crafting alternatives that do not require power tools. The icosahedron is one of the five Platonic solids. It has 20 faces, all of which are equilateral triangles.
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Template:
s5.postimg.org/qw8239447/golde...
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About my mathematical scribbles near the end:
Yes, I know that I used 'r' as a variable twice, but I ended up not using the 1/6 arc. In fact, the problem turned out to be really simple, and so I only used geometry, the Pythagorean theorem, and some ratios to answer my question.
See, math (as used in real life) is mostly just thinking out loud and stating the obvious, because that's what a systematic approach to an investigation does. It's like tire kicking, because you are uncertain about something, or else you wouldn't be doing a math problem in the first place; you are, after all, amidst a problem- by definition. Usually you are looking at some scratch sheet, and not at a finished piece of work. Yes, there are mostly redundancies all over the paper, but that's because I didn't know how to solve the problem at first, so I jumped right in by trying whatever popped into my head- and step one is to define the things I do know.
I often think of mathematics as like having another sense- like another way to see things. This sense can help us to gain a better conceptual understanding of the thing we are considering. Just as we have two eyes to give us depth perception, adding mathematical reasoning to the way that you look at the world will enhance the resolution of what you see, I promise.
Thanks for reading-
pocket83 
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Hope your head doesn't hurt from last night!
Here's some math to help soothe the pain. I also posted some additional content to this video on my 2nd channel. Happy new year ;)
pocket83² What's with the random nature of the 'dropped in' frames - I can see no pattern or correlation.
Can you use card instead?? doing it for school project
pocket83² where did you get '2a=a' and '2b=b'?
OK, I got everything except the rubber band part, which I thought was very complex and deserved a video on its own to show exactly how it was done.
:D
Not sure if it was intentional on your part, but this video was simplified to such an extent as to make it fairly difficult for anyone that has any woodworking experience to watch comfortably. I kept wanting to swat that utility knife out of your hand and march you over to the table saw :)
I think I lost all of the "woodworker" viewers. They don't take me seriously for some reason. But I do get lots of the
pocket83 The audience likes being fucked with occasionally. :)
I doubt I lost any subscribers worth keeping - if they can't take a little ribbing (or are too dense to see it for what it was), then they can step off. For every one that gets jettisoned, 100 more climb on to take their place.
I think we all have that same segment of the audience that doesn't have the tools or experience, but like to watch someone make something.
pocket83 I bet the followers of people like Chris Schwarz and Frank Howarth are captivated by your work. What you do is tactile and thought provoking. Woodworkers who care about proportions would do well to pay attention.
Pocket I do more "woodworking" projects, granted that I don't have too many tools. I think I was drawn to you by your "woodworking" projects, but your true audience is just entertained by your personality, creativity, and originality. I just hope that you don't steer away from woodworking projects because you don't think it fits your audience anymore. It fits your audience as long as you bring your values into that project.
@@pocket83 Just stumbling on this now. I'm slightly over the 25 demographic but since I'm a beginner, I really appreciated the simplification! :) Thank you!
Ahhh Geometry, the one thing in this world that can't be corrupted.
bill baggins
bill baggins tell that to Salvador Dalí
Ahhh Art. One of the most commonly corrupted things around.
bill baggins who corrupts them all? kicked out of every country multiple times. the that's the real puzzle. U know hew
House of Leaves
i never expected to learn something from a woodworking video about something i am really interested in, but the assembled icosahedron really amazed me. i'm probably going to dive into the maths behind this, thanks for the inspiration :)
I highly appreciated you walking me through your reasoning and mathematical analysis to just arrive at "less than 1/8" " I am really satisfied!
I absolutely love your stuff pocket, thanks again for the upload. I can't quite tell you exactly what it is, but your videos are extremely satisfying to watch. Cheers mate!
your projects are lots of fun to watch and build. some channels are fun to watch, some channels are only for building ideas. Pocket is good for both. thank you for taking the time to make these.
Nice video, very entertaining!
This is why I love both woodworking and math so much - together they can bring you loads of fun and some nice projects to do in your freetime.
Greetigs from germany,
Paul
Love the flower of life relationship. Perfectly fits when the lines are the proper width...
Your videos are always so dope dude, love your channel
2017 starts with a bang! Thanks for the great video. I'll be cutting those pieces out tonight to take into my classroom later this week.
Your videos are always amazing and truly interesting
Sometimes, you make me wish I had paid attention in geometry. I love it.
great video! i've been building icosahedrons big and small for over two years now. i did not realize they contained golden rectangles.
You always make the nicest projects. Keep it up 👍👍
I gave this one a like just for the narrative in your description. Well done.
I love building things out of that 5 mm underlayment! I keep a steady supply of it in my shop. I actually built my first delta style 3d printer from that exact stuff!
Pocket 83, you're such a cool guy and it's fun to hang out with you during your videos
Dissatisfied comment
I really enjoyed this video happy new year
Your presentation skills have really improved this last year, and in this video especially. Although I kind of thought the awkwardness in your old videos was sort of endearing, this is probably better.
I really like how you should the 3 planes that makes the shape. I did a project to make a 20 sided die to be 3D printed and came across this during research and thought it was remarkable.
Excellent. Math made fun! Thanks for the effort to put this together.
I just finished mine. Hope you might check it out. Setting up for the cut going through the center of each piece took way longer than I thought it would.
You really seem to like that sandpaper-on-a-stick.
Pocket! You're back!!!
hope your holidays went perfectly.
also a great project for the lasercutter, thanks to the template!
great video,very interesting subject and idea and uncommon,you have the admiration from a science teacher from Greece,!!!
That damned square root of 5 mucking things up.
I love how he has like 6 other ones in the background
Congrats on 100k subs!
Thanks for the book showing :) Just ordered
Pocket, I think you would be interested in "le modulor", it the golden ratio based on the human. Measurement which are comfortable for people.
i guess the number 1.732 approximates to √3, as the rectangle is made of two right angled triangles following the pythagorean triplet 1,2,and √3, not related to the golden section, but the six petal diagram can be worked out for 1.68...,
in fact there is a paper on this link based on floral geometry.
If you really want to, you can make the ratio using a compass, a pencil, and a straightedge. Use the pythagorean theorem with sides of 2 and 1, and you'll get sqrt (5) as a hypotenuse. With that, just add the length to the 1 and halve that total. length.
tiny, baby rubber bands on the ends of wires. then you can unfold flat pieces and put the rubber loops over tacks and make an antenna.
Very cool stuff.
Actually, you can construct a perfect golden rectangle using a compass since φ is constructible (you can get root 5 by constructing the hypotenuse of a 2 x 1 right triangle).
In fact, you can add, subtract, multiply, divide and take square roots of numbers with a compass and straightedge (you can even take cube roots if you cheat a little bit).
You're kinda right, and I knew that word would get scrutinized. But my point still stands on two technicalities: 1) the construction of an actual Fibonacci rectangle that's representative of the entire number would require infinite space and time, while we don't know if those variables are available to us, and 2) variation in any and every drawing implement yet devised introduces some random error, which prohibits perfection.
I love that mathematics creates these ideals for us to use as models, but it's important to remember that they can never be realized in the world in which we live. Political and economic theories are also tools that suffer the same limitations. This, in part, is why Plato considered his realm of 'forms' to be more substantive -and real- than our world.
One more interesting thing to consider: the arbitrary nature of irrational numbers. If our numerical system were based on a hypotenuse, rather than on a leg, root(2) would be a rational integer, and one would become an infinite decimal. In spite of Bertrand Russell's Principia Mathematica, it is nevertheless an assumption to consider one as a fundamental truth sufficient to build a mathematics predictive of all possible states of the universe. It may be that "perfect" eludes us because for us, perfect must be in terms of one, while the universe does not necessarily adhere to such a precept.
best d20 i've seen this week
Super video, thank you.
Nice video. Clear, entertaining (for those of a geometric mindset), and educational. An easier way to generate the Golden Number is Phi = 2 cos 36.
N914TJ Radian, or Degree?
The Dark_Speed_Ninja/ShadikkuX degree i belive
Ah platonic solids. I really enjoy these video's. Thanks for putting in the time and effort to build these. The copper one was beautiful top watch you make as well. Makes me want to build some rolling ball sculptures (namedropping David Morrell, specifically, his 66th build, the sphere), and add some wood elements to it.
Quick question; I seem to remember there being a system where you could use chopsticks and rubber bands alone to build stable solids, but it would only stay together once the tension was equal after adding the last rubber band. Something like "tensors"? It's kind of eluding me atm, and my google-fu is getting me nowhere.. Maybe you can shed some light on it?
Also, Those are some nice gears. Have you been hanging out with Izzy? :)
Also also, happy new year dude! Well wishings to you and your friends and family!
It's possible to make a "Hexastix" from pencils, which is a self-contained form that's pretty stable, if I remember correctly. That "stand-up math" guy made a video about it, I think (if you can tolerate his sarcastic anti-Americanism).
That shape could probably scratch the itch for you, but nothing else like that comes to mind right now. Thanks!
Hexastix, ah yeah I saw the video, but that wasn't what I meant. The rubber bands are not optional. I'm talking about making almost entirely hollow solids, but using rubber bands to pull the sticks through the center, or toward it, or into eachother's vertexes. No glue, no friction, just tension. I remember this one guy who was able to make stuff like rhombic cuboctahedrons, and icosahedrons, ,all the way down to simple cubes and tetrahedrons. It would be putty in his hands until he had the last string on. A very elegant form of tension based balance.
I don't mind the stand up maths guy though. I'm Dutch, so my tolerance for language based banter and patriotic jabs is quite up there :) But yeah, I learned to read, write and speak english from SKY Channel, when they still ran series like GI Joe, Mask, and transformers, so that might explain a thing or two :)
Cheers man, thanks for replying and thinking along!
Okay, okay, not 10 seconds after i posted above reply, I found it.
Tensegrity Structures/Models. YES! Wow, that's 10 years ago, and I (well, you and me both actually) had to dig deep to remember that. Wow.
Okay, well anyway, I'd love to see you build some stuff with this technique. There's a lot of disciplines and methodology there that reminded me of you. Damn I love it when a plan comes together!
/watch?v=8gtvxYZ0GIg
The ratio you obtained with the 6-point flower (1.7320508076) is actually sqrt(3) by the way.
Nice catch! I never recognized that!
This is pretty great to be honest.
Very nice work Pocket83! Excellent! However:
You may obtain a perfect Golden Rectangle by drawing a circle whose origin is the lower center of a square's lower edge, and whose radius is from the origin on through to the upper corner of the square. It is the point where the circle extends beyond the lower edge of the square that will define a perfect Golden Rectangle, being VOID of any approximations whatsoever. Have fun.
I should've kept my mouth shut there. I expected somebody to pick at that. Please don't take the following as an evasion, or as being smart-assed.
My point was more abstract than any approximate construction. "Perfect" is a theoretical ideal. Just like it's impossible to draw a perfect circle, the Golden rectangle can't be constructed because of its irrational nature. Even IF we had the ability to produce sides of zero thickness in the real world, the figure would still require an infinite space in which to draw it.
I have come to understand all irrational numbers as being descriptions of things that can't actually be defined within any numerical system that still assumes one as its fundamental. I have yet to be shown evidence to the contrary, but I certainly invite it. Our mathematics, much like the brains that invented it, are unfortunately quite limited. For example, we can make a symbol for infinity, so that we can use it operationally, but we can't know if infinity can even exist in reality. Such would be a (T)ruth claim, and those arguments are outside of the present scientific/philosophical purview.
Sorry if that all sounds fuzzy, but I would just admit it if I had misspoke in the video. I really _meant_ it when I said that drawing a perfect figure is impossible.
Even deeper:
The "perfect" description of infinity would be the set of all sets, even the set of all sets not possible, and of course, the set of all sets not possible become possible. In this place, I can't cease to exist, because my conscious experience reasserts its continuity immediately following my demise, because all cases are present. Whoa, is this the ontological argument? I'm not sure, but if infinity exists, we all come back again and again: we have to, because all sets, cases, scenarios, experiences, and meanings will exist an infinite number of times and in an infinite number of variations. Even the case where we can draw perfect figures will exist, which requires additional infinities.
Thus, I don't think infinity exists, and by extension, neither does perfect.
What are you looking for
I enjoy what some people might see as an "over-explanation". It's kind of the only reason I got through Algebra 3.
imagine the bottom line of a golden spiral as a movie timeline. would it mean that the best part of the movie is in the point where the spiral ends?
Who is going to have not one but two combination squares, but not a saw?
A kid who is not allowed to use the power tools in the garage. Not that it matters: you can make a marking gauge from a stick, so use two of those.
1 i dont have a garage or any tools.
pocket83 This is great, especially making it accessible to people without power tools. Also, phi pops up in the construction of a regular pentagon, which is my favorite way to get to it. (Also: ironically enough, I'd be more likely to let my six year old use a carefully set table saw instead of a box cutter on this project. It's a lot less likely to maim him. But not all parents are reasonable like that.)
Those are nice things that you bought
I'm looking for spicy Easter eggs
I can't believe you went through all that trouble just to find out it was just under 1/8"
the most badass rubber band ball in the world
I found what I was looking for.
so I did the math too, except I used some trigonometry instead.
I got b = a/tan(30 degrees), which, when we set a to 1, means b = 1.732050808..., which is the same as your answer.
1/tan (30) is equal to sqrt (3), which doesn't look that similar to (1 + sqrt (5))/2.
so yeah, it's just a coincidence that they look about the same.
I had that same bag of rubber bands. It took over 10 years for my house to use all of them. It probably takes you like 10 minutes on a project to do the same.
Re. the rectangle inscribed in the circle. You may appreciate how I saw the ratio "at a glance". The construction is simple.
imgur.com/a/qZcJK
Connect every corner of the rectangle to the center of the circle. Now all the short lines (red) are equal; They all equal the radius of the circle, r. So we have two equilateral triangles. The height of an equilateral triangle is root3 / 2 of the side. So the sum of the heights of both triangles is r*root3. And that is also the length of the rectangles longer side.
Hahaha :) loled at "What are you looking for" :)
No Pocket 83, what are you looking for in making these excellent videos?
I built a robot with the front, top and side views all using the eye-pleasing golden ratio in some manner. Then it turned out I screwed up measuring the batteries, and it needed a different set of sensors in front, and... sigh.
Cool video, you could have constructed the golden ratio geometrical too to save on mathing
Always interesting
Great! Tnaks for teaching and sharing.
The number you found at 11:20 is the square root of 3.
Dissatisfied comment: I appreciate your attention to detail and take offense to the suggestion that I wouldn't.
;-D
If you had access to a milling machine or router, could you possibly cut the slots in the middle out with a small enough bit?
Yes.
Luan is 3 ply underlayment.
9/10 IGN would be dissatisfied again
lovely
Do you Instagram?
Thank You for sharing your hardwork.
Apologies. Didn't mean it that way. Just seeing earlier comments made me wanna help clear things abit. Deleted my thoughtless comments. A very informative video actually. Thanks.
very kewl
You could also use foamcore instead of plywood
Definitely.
RawSauce338 or some mat board, like what thy use in picture frames. that stuff is great for architectural models
neat the video duration is 12:34
should have been 16:18
really nice vid! :D
Why do you add a sublim in the videos? And why didn't you complete the model with the band that would follow a wooden line?
hi which table saw do you use in the video. many thanks
What _am_ I looking for?
Wow!
5/64th is closer still. If you want to really get into it, 11/128th is wishing 6thiusandths of an inch
Amazing!!! Listening to you was like listening to Isaac Newton! If I was Bill Gates I will hire you right away! I 've created a board game buy yours made my jaw drop to the floor!
I love the sound of the tack being inserted into the wood. Is there something wrong with me?
Not at all. I go through lots of effort to give you that "feel like you're there" feel. I zoom in on stuff like that, and I appreciate that you notice!
Woo! Math!!!
very cool. I usually hear my art history professors refer to the Golden Number/Ratio as Phi. is there any difference there, between Phi and the Golden Number?
Nope. It's just a Greek letter, like Pi.
It just stands for the golden number.
The Golden Number, Phi, is also equal to 2 cos 36°.
Nice find. Messy, though. It requires that the calculator be set to Deg, not Rad. It's also conceptually tough; irrationality is not always easy to internalize, so removing it by more than one degree geometrically makes the expression become quite opaque to curiosity. I mean, I'm getting philosophical here, but what is 'cos θ,' really? It's just too arcane for me to understand. The Golden number here is being defined as something like, two of the quantity that's the ratio of two side lengths of the triangle that has an angle that's 1/10 of the circle. Whatever. Thanks for getting me to think this morning ;)
Satisfied comment.
That underlayment crap is 5 ply, 2 paper thin faces.
Yh ermm wow I really love the Golden but I struggle getting my head around the maths (yeah I am English I say maths it just doesn't sound right without the s)
a project that doesn't require a saw?
new year's resolution make something on pocket83's channel complete
maybe you've heard this before, but I think you would enjoy the "standupmaths" channel. he often makes weird geometrical thigngs.
Please use dark color rubber band
I'll just 3D print it
Isn't the platonic dodecahedron based on the silver rectangle the way a platonic icosahedron is based on the golden rectangle?
I wonder what polyhedra are based off the bronze mean and other metallic means.
en.wikipedia.org/wiki/Metallic_mean
What is the unit..? Cm ?
are u a math teacher with ocd obsession???
some how i feel very intrigue with all your videos... lols
new subs incoming. Have a nice day
I sense autistic vibes but idrk
Oh! I made one of these on a 3D printer, so I guess I made one without using a saw.
What brand of thumb tack are you using??
This is an example of dissatisfied comment
Its actually 8 and not 8.09... You can check it out in Fibo acci's Sequence.
The digits of the sequence are not exactly representative of the golden number. Kind of like how pi is not 3. As you add more digits, you get closer to it.
Just because two numbers belong to the sequence, that doesn't mean that they alone can define the golden number. Remember that it's irrational, so any definition of it that only uses integers will be an approximation.
if my memory serves me well, I detect an inside joke with where to put dissatisfied comments.
Wait nevermind. I wise slightly off.