Re: [Origami] Origami Sighting - Lexus Comercial

[Scott Cramer]

On Mon, Mar 26, 2018, at 4:04 PM, David Mitchell wrote:


> There are also the triakis polyhedra which can be derived through this 
> process, though they appear to be special instances

I believe the -kis polyhedra (triakis, tetrakis, pentakis) are derived by 
raising the centers of the faces of the original polyhedra so as to form 
triangular faces of a constant  dihedral angle. By contrast, some of the 
Archimedean solids have different dihedrals depending on which types of 
polygons meet at an edge.

The website http://dmccooey.com/polyhedra/ has interactive models of hundreds 
of polyhedra in amazingly abstruse categories, and exhaustive data on each. The 
interface could be a little more user friendly, but clicking on the boxes and 
playing around a bit is worth the effort. 

Scott 

Re: [Origami] Origami Sighting - Lexus Comercial

[David Mitchell]

From: Robert Lang  indited:

>In 1509, Luca Pacioli and Leonardo da Vinci put together a book, "La Divina 
>Proportione," in which they introduced the concept of erecting a pyramid on 
>each face of a >polyhedron (which is what is effectively happening in a Sonobe 
>solid) and they called the operation "Elevation." So Sonobe solids would be 
>(according to that terminology) >"elevated polyhedra," and that's the term 
>I've used when describing modulars with that form. Here's a Bridges paper that 
>gives details:

There do seem to be lots of names for similar aspects of this process. So ... 
as a non-mathematician I venture intrepidly into mathematical territory ... 
happy to have my misconceptions and mistakes pointed out.

At least the following possibilities seem to exist:

Elevation

Accretion

Cumulation

Akisation (from the Conway operator kis)

Kleetope

Pyramidisation or pyramid-augmentation

There are also the triakis polyhedra which can be derived through this process, 
though they appear to be special instances, as are those stellated polyhedra 
which can also be derived in this way.

George Hart suggests that elevation is also a special instance, the addition of 
pyramids with equilateral faces, but I do not know if this is an authoritative 
view.

I am not clear what the process of forming a Kleetope is and whether it is a 
specific process or a general one.

Of the remaining general terms I tend strongly towards cumulation because it 
means, or can mean, 'forming into a pile' which seems to make sense in terms of 
the process. 

Negative cumulation? Surely not? Excavation, perhaps?

And then, of course, I have myself been using pimpled and dimpled for years!

Dave

Re: [Origami] Origami Sighting - Lexus Comercial

[Meenakshi Mukerji]

>  http://archive.bridgesmathart.org/2014/bridges2014-235.pdf
>  http://www.georgehart.com/virtual-polyhedra/leonardo.html

Thanks for all your responses and resource pointers, Robert, Faye and
Francesco.

Elevated Polyhedra seems like a great nomenclature. I myself have
mistakenly called the Sonobe constructions stellated in the past but have
refrained from it more recently after learning the definition. Now I know
to call them elevated.

Regards,
Meenakshi

Re: [Origami] Origami Sighting - Lexus Comercial

[Eric Gjerde]

I’m not sure who did the Lexus commercial, but I had some conversations with 
them last year about wanting origami for some sort of commercial, so I think 
this is what they eventually went with. I’m curious to know who did it. It was 
a fairly complex and unusual brief.

As those intrepid origamists here who do commercial work can attest, the needs 
of advertising does not have a lot of room for purist origamist tendencies :) I 
don’t know how you folks manage to do it!

cheers

Eric

Re: [Origami] Origami Sighting - Lexus Comercial

[Dennis Walker]

My source for 'cumulation' was http://mathworld.wolfram.com/Cumulation.html

It even uses origami images, including a sonobe model!

Dennis

Re: [Origami] Origami Sighting - Lexus Comercial

[Dennis Walker]

Hi,

>>But then, I haven't found a good name for those Sonobe solids. How about 
>>pyramidized octahedron or icosahedron?

I don’t have an issue with calling them 'stellated' since it means 'turned into 
a star, but I'm aware that there is a strict geometrical definition of a 
stellation and I wondered about this some time ago.

The term I found was 'cumulation', so these would be cumulated solids. I also 
remember that it allowed for 'negative' cumulation, i.e. sunken faces.
I also found the word 'cacuminated' which means 'turned into a pyramid' but 
that was in a dictionary rather than a mathematical book.

(And to pre-empt any questions, I can't remember the source for 'cumulated'. 
I'll see if I can find it again!)

Dennis

Re: [Origami] Origami Sighting - Lexus Comercial

[Faye E Goldman]

Meenakshi:
A stellated octahedron can be thought of as a compound of two tetrahedra, where 
the spikes are smaller tetrahedra themselves ( 
https://en.wikipedia.org/wiki/Stellated_octahedron).  The Sonobe constructions 
have spikes that are pyramids bound by 3 right isosceles triangles and a 
triangular base. In the case of the Stellated Octahedron, the spikes are bound 
by 4 equilateral triangles. The Sonobe type constructions in no way satisfy the 
criteria for a stellated solid ( https://en.wikipedia.org/wiki/Stellation).

But then, I haven't found a good name for those Sonobe solids. How about 
pyramidized octahedron or icosahedron?

I've seen the terms 'elevated' or 'augmented' used to refer to these types of 
structures. The pyramid sticking out from the base polyhedron does not need to 
be anything special, unless you are looking for the stellation.
George Hart has some pictures from Leonardo Da Vinci's (1509)
http://www.georgehart.com/virtual-polyhedra/leonardo.html

Faye Goldman

Re: [Origami] Origami Sighting - Lexus Comercial

[Robert Lang]

Thus spake "Origami on behalf of Meenakshi Mukerji" 
 on 
3/19/18, 4:19 PM:

...The Sonobe type onstructions in no way satisfy the criteria for a 
stellated solid (https://en.wikipedia.org/wiki/Stellation).

But then, I haven't found a good name for those Sonobe solids. How about 
pyramidized octahedron or icosahedron?

In 1509, Luca Pacioli and Leonardo da Vinci put together a book, "La Divina 
Proportione," in which they introduced the concept of erecting a pyramid on 
each face of a polyhedron (which is what is effectively happening in a Sonobe 
solid) and they called the operation "Elevation." So Sonobe solids would be 
(according to that terminology) "elevated polyhedra," and that's the term I've 
used when describing modulars with that form. Here's a Bridges paper that gives 
details:

http://archive.bridgesmathart.org/2014/bridges2014-235.pdf

And if you web-search for "elevated polyhedron", you'll find other papers that 
use the term.

Robert

Re: [Origami] Origami Sighting - Lexus Comercial

[Francesco Mancini via Origami]

>But then, I haven't found a good name for those >Sonobe solids. How about
>pyramidized octahedron or icosahedron?

In italian we call them "ottaedro piramidato" and "icosaedro piramidato" that 
you can translate into the terms suggested by Meenakshi.

Ciao
Francesco 

"There's a fold in everything..." www.flickr.com/photos/mancinerie
 

Re: [Origami] Origami Sighting - Lexus Comercial

[Meenakshi Mukerji]

Thanks for both videos - the commercial as well as behind the scene. A
clarification about the shape. In origami we loosely call the Sonobe type
constructions stellated octahedron (12 units) or stellated icosahedron (30
units) while they are actually not. The the single sheet model in the ad
appears to have the same shape as a 12 unit Sonobe construction.

A stellated octahedron can be thought of as a compound of two tetrahedra,
where the spikes are smaller tetrahedra themselves (
https://en.wikipedia.org/wiki/Stellated_octahedron).  The Sonobe
constructions have spikes that are pyramids bound by 3 right isosceles
triangles and a triangular base. In the case of the Stellated Octahedron,
the spikes are bound by 4 equilateral triangles. The Sonobe type
constructions in no way satisfy the criteria for a stellated solid (
https://en.wikipedia.org/wiki/Stellation).

But then, I haven't found a good name for those Sonobe solids. How about
pyramidized octahedron or icosahedron?

Meenakshi

Re: [Origami] Origami Sighting - Lexus Comercial

[Metzger]

Thanks to Sarah and Peter for pointing out and sharing the Lexus stellated 
octahedron commercial:

https://lexusenthusiast.com/2018/03/08/lexus-celebrates-owners-with-new-commercial/

At first I thought, wow, that’s cool! Did they fold all those or was it just 
CGI? It was real, as can be seen by the “behind-the-scenes” video:

https://www.youtube.com/watch?v=xA18TAbnCYY

Yaacov