Zomadic Alchemy: The Triangle Question

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The Question

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Given a length of square stock and a standard compound miter chop-saw how would you make an equilateral triangle using only 45 degree cuts?
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 This is a woodworking math puzzle I invented. For three years now no one I've asked

has been able to answer the question.

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The Hints

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You need only set up the saw once

You may fashion a simple length stop
You will make exactly four cuts

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The Answer

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Set the saw to a 45 degree x 45 degree compound cut as shown.

Cut off the tail of the stock.

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Rotate the stock and off cut 180 degrees - this is critical.

partial rotation shown

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Rotate stock 180 degrees

completed rotation shown

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Use off cut as a length stop.

Slide stock to the stop.

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Make the second cut.

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remove first part

rotate stock 180 degrees and slide to stop

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With exactly four cuts you can generate three parts.

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Place the three parts flat on a surface

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Rotate the two side parts by 90 degrees

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Rotate one part around the fold axis.

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Fold the second part over by 90 degrees.

Here is an equilateral triangle made from only 45 degree cuts.

That's It!

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There is More!

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It's interesting that the two angled sides
of the triangle are not the same.
Depending on the respective orientation from these triangles 
one can make edge truncated octahedrons and tetrahedrons.

Consequently it follows that one can generate an octet-truss.

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Octahedron and Tetrahedron

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An octet-truss generated from a single part.

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Photos

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These alder frames are glued and nailed.

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Three triangles preparing to merge.

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The start of a tetrahedron.

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The start of an octahedron.

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Truss components

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PechaKucha

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My story starts here and it is a cautionary tale of what

 can happen when one stops watching television. 

In 2004 I built this structure called Spiky. 

It has an internal floor and you can climb in and sit down.

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Modular Origami and papercraft models are an excellent way
to learn about the geometry of space and nature. 

However the designer must be wary.
Paper can deceive.
Real materials, and real structures in the real world
have thickness which cannot be ignored.

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Just as the hand is an extension of the mind,
so are tools extensions of the hand.

Today, access to a quality cnc router with a skilled operator
is one of the most powerful tools a design/builder can have.

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Here we see expressions in wood exploring joinery systems and
different types of architectural primitives and design elements.
Well designed complexity is the result of compositions of patterns
each of which can be the epitome of simplicity.

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For three years I engaged in full time commercial cnc millwork and furniture fabrication.

Fabricating the designs of others gave me some of 

the experience and skills I now have to effectively realize my own designs.

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My aesthetic emphasizes simplicity and ease of construction. 

I strive to create fixtures and spaces which engender
a sense of embrace and are drawn from living patterns
which are mentally healthy and domestically efficacious.

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Modern fabrication technology enables us
all to break with convention.
Here, instead of buying a christmas tree,
or buying a fake tree,
I've designed my own holiday scupltural icon.
And the beauty of this tree is that
it is it's own ornament.
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In my research of architectural and generative design systems 

I discovered a wonderful design pattern called the Zome;

a structure derived from polar zonohedra. 

Zomes are very modular and flexible such that 

the same panels can be recombined in a  

wonderful and bewildering variety of ways.

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The Zomicile is my first full scale shelter structure.
Composed of only five unique parts
it assembles using only a mallet.
It's amazingly strong and is designed to shed rain
with it's series of overlapping panels.
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My second Zome structure is a greenhouse for an urban garden.

Here we see the painted panel frames awaiting
the exterior grade plastic sheeting each pair will sandwich.

The color resonates with the purple onions growing in the garden.

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This Zome is  25' long, 10' wide and 10' tall.
The prefabricated parts were assembled in a single day by
a small group of friends with little or no construction experience.

This time every panel was tongue and grooved to it's neighbors.

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Now let's all gather round and discuss somethings I've learned. 

Beautiful well designed shelter
does not have to be a high end commodity. 

Aesthetics are important and are not purely subjective. 

And manageable complexity can be generated from simplicity.

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Before committing time and materials to a full scale project
it's almost alway worthwhile to test build small scale designs
of the various elements which will compose the final structure.

Here we see some of the latest manipulatives I've designed
to help me explore the design possibilities of this architectural pattern.

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In terms of simplicity, modularity and flexible design  

power Zomes are second to none.

What of the cube you might ask?  

Surely the cube is king. 

Well, my friend, the cube is a Zome.
It is the simplest Zome of all.

And therein lies it's power.

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This project explores the minimal structure of a Zome. 

I wanted to build a space which would  intrigue
the participant from near and far. 

To create the suggestion of a space
which brings the outside in and lets the inside out.

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Here are all the joinery elements.

This structure is completely homogeneous. 

A single tool cut all the parts. 

A single material was used in the fabrication. 

And it can be assembled by anyone  

with their bare hands.

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While not always obvious, a truism I've learned is that all systems are polyhedra.

And they divide the Universe into two parts
...the inside and the outside.

And good design creates effortless transitions between the two.

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This is called the Bodhisattva Zome.

And as that name indicates it implies an intention of compassion
and a desire to share the knowledge I’ve learned
and the skills I’ve gathered with all people to whom it would benefit.

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There's more to good structure
than modern sophisticated engineering.

There's more to good design
than clever and impressive tricks.

In all aspects of our creative endeavors...

Only the best is good enough.

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As you view this manifestation of my imagination

and all that it implies, you may note that it is seemingly

only half there and half not.

Is it fading away or coming into being?

Which of those two possibilities would you prefer it to be?