2:19 Correction: The vertices in both rings are the same, but the "polar" vertices (two squares) are distinct from the "equatorial" vertices (two octagonal rings)
Over quarantine summer I started to make the archimedean solids out of paper and masking tape, but sadly I only finished 6 of the 13 before I ran out of steam, energy, and concentration. I ended up moving onto other hobbies such as painting and biking, but I wish my ADHD had let me stick to these.
I didn't quite follow the hand-wavy bit about how one gets around the problem where truncation of certain vertices would seem to yield rectangles, but "done right" it yields squares.
Okay, so truncate the cuboctahedron such that you do wind up with those rectangles. Now consider that version of the rhombicuboctahedron, centered at the origin and oriented such that the six squares are parallel to the XY, YZ, and XZ planes. Next, note that the set of edges that pass through the XY plane are the short edges of some of the rectangles, and note that they are all vertical. Since they are all parallel to each other, you can elongate all of them without disrupting any of the angles or other edges; so in particular you can elongate them just the right amount to make the rectangles into squares. Doing so turns the original squares into rectangles, but then doing the same for the YZ and XZ planes fixes those, so all the original squares now have edge length equal to the long edges of the rectangles, and the rectangles are all squares. Not sure how helpful that explanation is without any visual aid though...
Great summary, and nice printouts! Follow-up question: if the rhombicuboctahedron has a variant that can be made by rotating one of its square cupolas (the elongated square gyrobicupola), what about the rhombicosadodecahedron and rotating one (or more) of its pentagonal cupolas?
Good point Matt. The family of gyrate rhombicosidodecahedra (rhombicosododecahedra with one or more rotundae rotated by an edge) each have some pairs of adjacent squares. In that way, they are like the triangular orthobicupola
Personally I'm more into the pontoremeyotoqatonodexofantopoloyotrahedron. Easy to get mixed with the rotlokoizooxloqatolatoronopoloitonosomonidolopilokalokanoyotctafantohedron.
Traduje tu comentario a inglés. "Please translate to Spanish how to make these figures or what machine is used; let me know where to get these in Colombia."
Only because they are explicitly ruled out to get a finite collection. There's more of a debate over why the en.wikipedia.org/wiki/Elongated_square_gyrobicupola is not included.
Thanks to the life work of Frank Chester many more forms have been discovered... The 7 sided Chestehedron is but one of his path finding discoveries. He also gave us the method to unveil the complete family of complex solid forms. All forms exist between the Sphere & Cube
Can a sphere be described as a regular solid with an infinite number of faces? I know that it is problematic because a circle can be described as a regular polygon with an infinite number of sides also. So if the circle already has an infinite number of sides, how can a sphere be constructed? Also, How can there be more than one circle? How can it even be observed? It can be constructed with Euclidian mathematics yet cannot be adequately defined. Just wondering? I am pretty sure this is why the value of pi cannot be precisely calculated.
You could say that a sphere is the _limiting shape_ of a certain sequence of regular polyhedra (and to be pedantic, we have to be careful about what kind of limit we are taking). The circle can certainly be defined - it is the set of points at a given distance from some center point. And pi can be precisely calculated - take the ratio of the length of the circumference to the length of the diameter. You are right that we cannot write pi down as a decimal expansion of finite length, but that doesn't mean that we don't know precisely what it is.
A lovely set of models. Thank you.
The truncated icosahedron as you know is also the skeleton of the peculiar carbon molecule buckminsterfullerene.
s o c k e r b o l l
Lol
2:19 Correction: The vertices in both rings are the same, but the "polar" vertices (two squares) are distinct from the "equatorial" vertices (two octagonal rings)
Over quarantine summer I started to make the archimedean solids out of paper and masking tape, but sadly I only finished 6 of the 13 before I ran out of steam, energy, and concentration. I ended up moving onto other hobbies such as painting and biking, but I wish my ADHD had let me stick to these.
Ik how to make those out of paper but masking tape ? Did u use them to stick it or something else because I am a little confused
This was great to watch, the models are lovely
Platonic and Archimedean solids.
I didn't quite follow the hand-wavy bit about how one gets around the problem where truncation of certain vertices would seem to yield rectangles, but "done right" it yields squares.
Okay, so truncate the cuboctahedron such that you do wind up with those rectangles. Now consider that version of the rhombicuboctahedron, centered at the origin and oriented such that the six squares are parallel to the XY, YZ, and XZ planes.
Next, note that the set of edges that pass through the XY plane are the short edges of some of the rectangles, and note that they are all vertical. Since they are all parallel to each other, you can elongate all of them without disrupting any of the angles or other edges; so in particular you can elongate them just the right amount to make the rectangles into squares. Doing so turns the original squares into rectangles, but then doing the same for the YZ and XZ planes fixes those, so all the original squares now have edge length equal to the long edges of the rectangles, and the rectangles are all squares.
Not sure how helpful that explanation is without any visual aid though...
Make the Johnson solids! Not all of them, only the ones listed under "others" on Wikipedia
Here you go! shpws.me/Hmb8
Wow, that was fast. Thanks!
What is the platonic and Archimedean solid that can be made using the greatest amount of faces, forming a near-spherical shape?
snub dodecahedron
my favorite group of5 and the rest of the 13 arc solids and the 13 cat solids and the 92 johnson solids
Great summary, and nice printouts!
Follow-up question: if the rhombicuboctahedron has a variant that can be made by rotating one of its square cupolas (the elongated square gyrobicupola), what about the rhombicosadodecahedron and rotating one (or more) of its pentagonal cupolas?
Yes, those are Johnson solids. See the Gyrate rhombicosidodecahedron for example.
That solid exists but it is not as tricky a challenge to the definition of an Archimedean solid, because its vertices are not locally identical.
Good point Matt. The family of gyrate rhombicosidodecahedra (rhombicosododecahedra with one or more rotundae rotated by an edge) each have some pairs of adjacent squares. In that way, they are like the triangular orthobicupola
beautiful presentation!
Make the 92 Johnson Solids next!
And the 13 Catalan solids (duals of Archimedean solids)
The rhombicosidodecahedron is my favorite shape, its the perfect mix of round and blocky
Personally I'm more into the pontoremeyotoqatonodexofantopoloyotrahedron. Easy to get mixed with the rotlokoizooxloqatolatoronopoloitonosomonidolopilokalokanoyotctafantohedron.
@@deadzoneRL-q3v fanum tax?
@@deadzoneRL-q3vidk man .. if those words are real, then Terrence Howard is onto something. Damn those Anunnaki and straight lines
6:37 me trying to draw a soccer ball (football) but I keep forgetting what shape the pieces are
Would there be any more shapes possible if the regular polygon faces requirement was removed, but still needed all vertices be the same?
Would you kindly give an introduction to Aristotle s philosophy behind the solids of his imagination...what was he thinking about them?
These are great looking models I'd love to know how you made them? Did you create moulds?
They are 3D printed, in selective laser sintered nylon.
Where can I get a set if these polys? Great video thanks.
i do like these, but personally much more of a fan of the duals to the archemedian solids, the catalan solids.
Where can I get a set of these polyhedra? Great Video thankyou.
Links in the video description!
excellent
Have you seen the truncated rhombicosidodecahedron?
Alright this is cool shit. Subscribed.
all i know is the tetrahedron " structurally speaking " outlasts them all
6:35 That’s a classic soccer ball!
traduzca en español como elaboro esas figuritas o que maquina se utiliza por favor informe donde las consigo en colombia
Traduje tu comentario a inglés.
"Please translate to Spanish how to make these figures or what machine is used; let me know where to get these in Colombia."
good video ..I'd like to have those shapes ..geometry is good vibes
Why don't the infinite number of prisms and antiprisms fit in here?
Only because they are explicitly ruled out to get a finite collection. There's more of a debate over why the en.wikipedia.org/wiki/Elongated_square_gyrobicupola is not included.
are platonics part of the archimedians?
Usually the platonics are not included in the archimedians, but it depends on your definitions.
from where can i buy these cool looking solids ?
Links are in the video description.
use polyhedrons to stack them
Thanks to the life work of Frank Chester many more forms have been discovered...
The 7 sided Chestehedron is but one of his path finding discoveries.
He also gave us the method to unveil the complete family of complex solid forms.
All forms exist between the Sphere & Cube
Unfortunately, the quadrilaterals of the Chestahedron are not regular, and so it is not an Archimedean solid.
Elongated square bicupola? Is rhombic cuboctahedron
Has anyone tried to make 4 dimensional Archimedian solids?
idk look it up on wikipedia or smth seriously there’s a whole page there
😍😍😍
I can tell that this video is very informative, but I’m having a hard time understanding this haha
what type of 3d printer are you using?
austin collier I use shapeways.com's "White Strong & Flexible" material, which is a nylon plastic printed using a selective laser sintering machine.
Can a sphere be described as a regular solid with an infinite number of faces?
I know that it is problematic because a circle can be described as a regular polygon with an infinite number of sides also. So if the circle already has an infinite number of sides, how can a sphere be constructed? Also, How can there be more than one circle? How can it even be observed? It can be constructed with Euclidian mathematics yet cannot be adequately defined. Just wondering? I am pretty sure this is why the value of pi cannot be precisely calculated.
You could say that a sphere is the _limiting shape_ of a certain sequence of regular polyhedra (and to be pedantic, we have to be careful about what kind of limit we are taking).
The circle can certainly be defined - it is the set of points at a given distance from some center point. And pi can be precisely calculated - take the ratio of the length of the circumference to the length of the diameter. You are right that we cannot write pi down as a decimal expansion of finite length, but that doesn't mean that we don't know precisely what it is.
Something Something Muoctahedron
Truncated cuboctahedron = great rhombicuboctahedron
Truncated icosidodecahedron = great rhombicosidodecahedron
(topologically)
Gimmie Catalan solids
Truncated icosahedron is soccer ball
rhombihedron
Who came here after spiderman no way home??
Julia Quinn's
JQ
test
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