Rhombicuboctahedron curler units

Designer: Herman van Goubergen

Reference:  http://www.youtube.com/watch?v=TVtCcinsP9c&list=PL83B4618216D7989B by Jo Nakashima

No of units:  24 units.

Paper size used : 3" x 3"  

Difficulty: easy to make units and simple assembly.

Paper used: normal color printer paper

Time: 1 hour

Rhombicuboctahedron is an Archimedian solid with 8 triangular and 18 square faces. 
Further reading: http://en.wikipedia.org/wiki/Rhombicuboctahedron

Icosidodrcahedron - Rhombicuboctahedron - Cuboctahedron

kusudama versailles

Designed by Krystyna Burczyk

Presenter: http://www.youtube.com/watch?v=RmNeREucbIg by Tadashi Mori

Paper used : Color printer paper

Paper size:  4 " x 4 "

Single flower with single unit

Difficulty: Easy, assembly is easy.

No of units: 60 to make a kusudama ball. 5 units to make one flower. 12 flowers required totally.

12 flowers

Time : I took 5 days. I would have taken 6-7 hours totally to make 60 units and assemble.

No tools used for curling, no glue used for assembly.

Single unit with Completed Kusudama ball

It is easy to make units and assemble. Didn't find any problem. It's a beautiful bunch of paper flowers.It looks more beautiful in real than in photo.



Interesting math:

When we fold the first two steps, we actually dividing 90 degrees into three 30 degree angle. Simple proof using Trigonometry:

 θ = 60 degrees.
 α = θ / 2 = 30 degrees. (α is angular bisector of θ)

I have used 4 inch square sheet. Side of square would be 4 inch. AB is 4 inch since we fold the tip of below right corner of square, let's say D, to point A. Hence AB = BD = 4 inch.

Started on 17th June 2013. Completed on 21 st June 2013

Archimedean solids

When I did Icosidodecahedron curler unit, I went curious about this polyhedron and searched on the net. I found some really interesting facts I thought worth documenting here for quick reference.

 Archimedean solid, (From Wiki) it is a highly symmetric, semi-regular convex polyhedron composed of two or more types of regular polygons meeting in identical vertices. They are distinct from the Platonic solids, which are composed of only one type of polygon meeting in identical vertices.

Icosidodecahedron and Cuboctahedron are one of the 13 Archimedean solids. For more refer to this link: http://www.geom.uiuc.edu/~sudzi/polyhedra/archimedean.html

There are 5 (and only five) Platonic solids namely Cube, Tetrahedron, Octahedron, Dodecahedron and Icosahedron.

 Truncation means cutting off the corners of a solid. We cut off identical lengths along each edge emerging from a vertex. This process adds a new face to the polyhedron.

What happens when we truncate Icosahedron?

If, when truncating the vertices of either the icosahedron or the dodecahedron, we move in exactly half way, the result is the icosidodecahedron. Thus, we can think of the icosidodecahedron as being the limiting case of either the truncated icosahedron or the truncated dodecahedron.

There's another Archimedean solid worth mentioning. That is Truncated Icosahedron. (polyhedron in Ten intersecting Planes)While truncating Icosahedron if we move 1/3rd we get Truncated Icosahedron. A good example of this solid is a Football. 

And if we move 1/2 we get Icosidodecahedron.

There's a reason why we arrive at Icosidodecahedron when we truncate either Icosahedron or Dodecahedron half way. The reason is they are Dual Solids where the vertices of one corresponds to faces of other. 

If we take Platonic Solids:

1) Dodecahedron and Icosahedron are dual solids.

    Explanation: Dodecahedron has 12 faces and 20 vertices

                       Icosahedron has 20 faces and 12 vertices

2) Cube and Octahedron are dual solids.

     Explanation: Cube has 6 faces and 8 vertices

                         Octahedron has 8 faces and 6 vertices.

3)  Tetrahedron is self dual. It has 4 faces and 4 vertices.

Refer to my Origami platonic pictures to verify the above fact. 

Some references:

http://www.geom.uiuc.edu/~sudzi/polyhedra/  - In this site, under Archimedean solids, there's lovely illustration of diagrams to help understand better. 

http://en.wikipedia.org/wiki/Icosidodecahedron

http://en.wikipedia.org/wiki/Archimedean_solid

http://en.wikipedia.org/wiki/Dual_polyhedron

http://www.coolmath.com/reference/polyhedra-truncated-icosahedron.html

Icosidodecahedron Curler Units

I loved cuboctahedron curler unit and got inspired to try Icosidodecahedron curler unit. Making single unit and assembly is same as Cuboctahedron.
When I first heard "Icosidodecahedron" I thought it is some complex module(the complicated name mislead me), after I completed I felt that this is easier than Cuboctahedron. In Cuboctahedron pieces kept coming out as we need to stretch paper in final assembly. In this one no such hassle, the units are easy and simple to assemble.

I had earlier done Icosahedron but this one is different. When I googled about Icosidodecahedron I learnt that it is a polyhedron which has 12 pentagons and 20 triangles. Each pentagon is surrounded by triangles on each of it's five sides and each triangle is surrounded by pentagon on each of it's three sides. Remember this line when assembling, nothing will go wrong and it's easier this way instead of following video. I watched the videos once and assembled myself easily.

In the above picture notice the pentagon formed by green, pink, yellow, orange, blue papers. Also notice the triangles surrounding this pentagon.

Further I was curious to know about this polyhedron, I made some search on the net. I found some really fascinating facts. I will post it in a separate post.

Cuboctahedron and Icosidodecahedron

Reference:  Designer: Herman van Goubergen: http://www.britishorigami.info/academic/curler.php
                  Presenter: http://www.youtube.com/watch?v=e5V2GzCxMXs by jonakashima
                                  http://www.youtube.com/watch?v=WoRKoXm-4rI  by barbabellaatje

No of units: (just) 30 units.

Paper size used : 3" x 3"  

Difficulty: easy to make units and simple assembly.

Paper used: normal color printer paper

Time: 2-3 hours

Started and Finished on 15 May 2013

Cuboctahedron Curler Units

Designed by Herman van Goubergen

Reference:  Designer: Herman van Goubergen: http://www.britishorigami.info/academic/curler.php

                  Presenter: barbabellaatje     : http://www.youtube.com/watch?v=lm2JEhBvxCg 

No of units: (just) 12 units.

Paper size used : 4" x 4"  

Difficulty: easy to make units and simple assembly.

Paper used: normal color printer paper

Time: 2-3 hours

Cuboctahedron is a polyhedron with eight triangular faces and six square faces. 

Assembling technique: Follow : In a Cuboctahedron triangles are surrounded by 3 squares and all squares are surrounded by 4 triangles. 4 curls meeting forms a square, 3 curls meeting forms a triangle.

Single Curl and finished Cuboctahedron

I felt the units are very easy to make and assembly is simple. This is the first curler model I did. It took me a while to get the curls properly, after curling about 2 units, I got the hang of it. This model is based on water bomb base. Most of the curler models requires paper to be curled and these curls holds each other. There's no flap or pocket. No need of glue. 

While assembling this I stumbled twice. Both the times the units came apart at the last step. To reassemble it was confusing, I had to dismantle again and start from the beginning. 

I realized after assembling I had tightly curled the units so they don't break, this was the problem, I could not stretch the paper units because it was tightly assembled and when I forced to bend everything broke.

Third time I tucked in the papers but loose enough so I could stretch the units into a ball while joining the last step. Once assembled the units sit tight. (I have even dropped this model many times while moving here and there, but it didn't break.)

A beautiful, easy and simple curler unit to start with. I did not use any tool to curl, no glue and no paperclip while assembling, just my thumb and index finger :)

Started and finished on 13 May 2013.