The Perfect Equalateral Triangle

[Paul Kranz]

This would be a riddle except I don't know the answer: What are the fewest cuts you can make to a 4 X 8 sheet of plywood (32 sq. ft.) that would cover a 32 sq. ft. equilateral triangle? Assume no wood is lost by the blade.

For these purposes let's define the perfect equilateral triangle as having a side of  sqr(32 / sqr(3)) * 2 or 8.59656 ft. or 8'7-5/32".

I am trying to figure out a way to utilize an entire sheet of plywood for one equilateral triangle while minimizing the number of pieces of plywood.

P.S. Did anyone catch Dustin Feider on that Animal Planet Treehouse show?

Paul sends...

[Thomas Rainwater]

Interesting! I have no idea, but I'm immediately reminded of the Haberdasher's problem. (And Dudney's solution.) Might be a seed crystal for thought....

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

-Tommy Rainwater

-- 
http://www.thomasrainwater.com
thomas.r...@gmail.com

--
--
You received this message because you are subscribed to the "Geodesic Help" Google Group
--
To unsubscribe from this group, send email to GeodesicHelp...@googlegroups.com
--
To post to this group, send email to geodes...@googlegroups.com
--
For more options, visit http://groups.google.com/group/geodesichelp?hl=en

---
You received this message because you are subscribed to the Google Groups "Geodesic Help Group" group.
To unsubscribe from this group and stop receiving emails from it, send an email to geodesichelp...@googlegroups.com.
For more options, visit https://groups.google.com/d/optout.

[Paul Kranz]

Thomas:

Great resource! Thank you very much. If it can be done with a square, maybe it could also be done with a rectangle (double-square).

Paul sends...

--

Very high regards,

 

Paul sends...

ReviveTheFlame.com

[Gerry in Quebec]

Hi Paul,

The attached jpg doesn't give the precise answer you're looking for, but it may be of use. The saw kerf/sawdust would account for about half the 1/4"-wide remainder strip.

- Gerry in Quebec

[Hector Alfredo Hernández Hdez.]

I like so much :)

--

[Gerry in Quebec]

One way to make full use of 4'x8' plywood or other similar sheathing for a dome is to use triangle shapes that naturally lend themselves to subdivision from a rectangle. This requires a bit of exploratory geometry. The attached jpg is an example of a "plywood-perfect" approach to dome design. It shows an adapted octahedral hemisphere (class I, frequency 3). With this arrangement, the dome has a floor diameter of just over 25 ft, 3 types of triangles, and 3 distinct radii. Of the 32 triangles, 28 are plywood-perfect, meaning each can be covered with exactly one 4'x8' sheet.

- Gerry in Quebec

On Wednesday, August 20, 2014 9:23:32 PM UTC-4, Paul Kranz wrote:

[Adrian Rossiter]

Hi Paul

On Thu, 21 Aug 2014, Paul Kranz wrote:
> Great resource! Thank you very much. If it can be done with a square, maybe
> it could also be done with a rectangle (double-square).

Here is a solution with seven cuts, as an initial number to beat.

The red line is the same length as the short side, and the angles
at the pink dots are 60 degrees.

Adrian
--
Adrian Rossiter
adr...@antiprism.com
http://antiprism.com/adrian

[Gerry in Quebec]

Hi Adrian,

How about giving us a hint how to arrange the 8 pieces into an equilateral triangle? I tried assembling paper cutouts using your image but couldn't get it to work. And when I calculated the segment lengths in your rectangle using a width of 4 and a length of 8, I couldn't get any combination of them to add up to the edge length of Paul's equilateral triangle, namely 8.59656 feet. Thanks.

- Gerry

[Adrian Rossiter]

Hi Gerry

On Sat, 23 Aug 2014, Gerry in Quebec wrote:
> How about giving us a hint how to arrange the 8 pieces into an equilateral
> triangle? I tried assembling paper cutouts using your image but couldn't
> get it to work. And when I calculated the segment lengths in your rectangle
> using a width of 4 and a length of 8, I couldn't get any combination of
> them to add up to the edge length of Paul's equilateral triangle,
> namely 8.59656 feet. Thanks.

My image is incorrect. The idea is just a simple rearrangement of
the solution to Haberdasher's problem. Unfortunately I was looking
at version of the dissection where the final triangle wasn't
equilateral!

The correct rearrangement still has seven cuts. In the attached
image translate the right triangle to the left side, and the
bottom triangle to the top side, to make the 1x2 rectangle.

Adrian.

[Gerry in Quebec]

Hi Paul & others interested in efficient sheathing,
To follow up on the topic of making good use of 4'x8' sheets for building domes and other triangulated structures, here's another example of a "plywood-perfect" design. In the attached jpg, the polyhedron (left) has 24 faces, all identical isosceles triangles. When the radius to the highest vertex and lowest vertex is 0.8920, the radius to the remaining 12 "nonpolar" vertices (the spherical radius) is 1. When the spherical radius of the polyhedron is 8.9282 ft, each triangle can be covered with exactly one 4'x8' sheet. This is done by ripping the sheet on the diagonal, corner to corner, to create two right-angled triangles.

If you make a "dome" out of this polyhedron by removing the bottom 6 triangles, the footprint is a regular hexagon with 8 ft edges (footprint diameter = 16 ft). Total floor area: 166.2769 sq. ft. Dome height: 11.9279 ft.

- Gerry in Quebec

[Paul Kranz]

Here are 8 shapes that make a 32 sq ft rectangle, square and equilateral triangle.

Paul sends...

--
--
You received this message because you are subscribed to the "Geodesic Help" Google Group
--
To unsubscribe from this group, send email to GeodesicHelp...@googlegroups.com
--
To post to this group, send email to geodes...@googlegroups.com
--
For more options, visit http://groups.google.com/group/geodesichelp?hl=en

---
You received this message because you are subscribed to the Google Groups "Geodesic Help Group" group.
To unsubscribe from this group and stop receiving emails from it, send an email to geodesichelp...@googlegroups.com.
For more options, visit https://groups.google.com/d/optout.

[TaffGoch]

Here's my quick rendition:

The real question is "how few" pieces (or fewest cuts".) There are many possible solutions, the most-elegant being those with the fewest pieces.

How it's done:

The general method for solving this type of dissection "puzzle" involves "grid" overlay of the two geometric shapes that you are trying to dissect, making sure that the two shapes have the same surface area.The solution method for the square-to-triangle dissection looks like this:

This solution is so elegant because several edge midpoints coincide, square and triangle.

My rectangle-to-triangle solution started with this grid overlay:

(Note that, to begin, I drew both the rectangle and triangle to have the same surface area.)

I only played with this a little while, so I'm sure that a more-elegant solution likely exists, with fewer than 6 pieces. More experimentation with rotation angle, translations and midpoint coincidence should/could produce a more-pleasing dissection. This method description should give you a good starting point, if you want to continue your search for a nice solution.

-Taff

[TaffGoch]

By the way, the above dissection method is described in the book:

A reference to the method appears at the webpage:

http://www.takayaiwamoto.com/Dissection_Tesselation/Dissection_Tesselation.html#tr2sqr

Scroll down to this image (and description):

"...standard procedure used in finding the dissection from one geometric shape to another:

(1) arrange triangles and squares horizontally. 

(2) move the group of triangles such that mid-points t4 and s5 coincide and t3 falls on the line s7-s8.

The location of this point can be computed by: 

Pythagorean theorem : (t3 - s8)² = (t3 - t4)² - (s5 - s8)² "

_________________________

Method animated (from above-referenced webpage):

Note that mid-point coincidence is, apparently, NOT a requirement, as depicted in a different dissection problem:

-Taff

[Paul Kranz]

Taff:

Will those same six shapes form a square?

Paul sends...

--

--
You received this message because you are subscribed to the "Geodesic Help" Google Group
--
To unsubscribe from this group, send email to GeodesicHelp...@googlegroups.com
--
To post to this group, send email to geodes...@googlegroups.com
--
For more options, visit http://groups.google.com/group/geodesichelp?hl=en

---
You received this message because you are subscribed to the Google Groups "Geodesic Help Group" group.
To unsubscribe from this group and stop receiving emails from it, send an email to geodesichelp...@googlegroups.com.
For more options, visit https://groups.google.com/d/optout.

[TaffGoch]

Haven't tried square fitment. I was working on the problem posed in the original posting.

-Taff

[Paul Kranz]

Taff:

Six shapes is better than eight. Thanks for the input.

Paul sends...

On Sat, Aug 30, 2014 at 12:26 AM, TaffGoch <taff...@gmail.com> wrote:

Haven't tried square fitment. I was working on the problem posed in the original posting.

-Taff

--
--
You received this message because you are subscribed to the "Geodesic Help" Google Group
--
To unsubscribe from this group, send email to GeodesicHelp...@googlegroups.com
--
To post to this group, send email to geodes...@googlegroups.com
--
For more options, visit http://groups.google.com/group/geodesichelp?hl=en

---
You received this message because you are subscribed to the Google Groups "Geodesic Help Group" group.
To unsubscribe from this group and stop receiving emails from it, send an email to geodesichelp...@googlegroups.com.
For more options, visit https://groups.google.com/d/optout.

[norm...@gmail.com]

I've been following this post with interest.  My curiosity forces me to ask:  Is this more of a thought experiment or do you plan on doing this in real life?

[Paul Kranz]

Normalson:

It is quite practical. If I can figure out how to utilize all of the plywood, I can construct a bigger dome with the materials at hand making dome building more economical.

I am working on a computer program that will crawl around on a 4 X 8 sheet of plywood to see if there is a way to orient an over-sized intersection of three 60-degree angles to find out how close I can get to perfection (re: Taff's contribution to this thread). Computers are great for this kind of work.

If the perimeter of the perfect equalateral triangle is 6 * sqr(32 / sqr(3)), then the total length of the lines on the plywood heading away from the 60-degree angles must be the same or there is no need to further evaluate the position or orientation of the intersection on the plywood.

I will let you all know what becomes of it.

High regards,

Paul sends... 

On Thu, Sep 4, 2014 at 9:45 AM, <norm...@gmail.com> wrote:

I've been following this post with interest.  My curiosity forces me to ask:  Is this more of a thought experiment or do you plan on doing this in real life?

--
--
You received this message because you are subscribed to the "Geodesic Help" Google Group
--
To unsubscribe from this group, send email to GeodesicHelp...@googlegroups.com
--
To post to this group, send email to geodes...@googlegroups.com
--
For more options, visit http://groups.google.com/group/geodesichelp?hl=en

---
You received this message because you are subscribed to the Google Groups "Geodesic Help Group" group.
To unsubscribe from this group and stop receiving emails from it, send an email to geodesichelp...@googlegroups.com.
For more options, visit https://groups.google.com/d/optout.

[Gerry in Quebec]

Hi Paul,

The more pieces of sheathing you have to cut to cover a given triangle, the longer it takes and the greater the likelihood of cutting errors, which would increase the amount of plywood needed. There's also the issue of having to use more backers, namely wooden braces within each triangle to support multiple plywood subpanels. This will undermine any savings achieved by zero waste of plywood. It will also increase thermal bridging.

 

The division of a rectangle whose length is twice its width into a minimal number of pieces that exactly cover an equilateral triangle of the same area is an interesting armchair exercise. But I don't think the results would be very practical for sheathing a triangulated dome with 4x8 sheets.

- Gerry

[Paul Kranz]

Gerry:

I can appreciate that. I will be more convinced after I have determined how goofy the cuts have to be to get it right. What might be a more economical use of materials may be undermined by over engineering. At this juncture, I have the time for the exercise.

Paul sends...

--
--
You received this message because you are subscribed to the "Geodesic Help" Google Group
--
To unsubscribe from this group, send email to GeodesicHelp...@googlegroups.com
--
To post to this group, send email to geodes...@googlegroups.com
--
For more options, visit http://groups.google.com/group/geodesichelp?hl=en

---
You received this message because you are subscribed to the Google Groups "Geodesic Help Group" group.
To unsubscribe from this group and stop receiving emails from it, send an email to geodesichelp...@googlegroups.com.
For more options, visit https://groups.google.com/d/optout.

[TaffGoch]

...and there's "the thrill of the hunt!"

[norm...@gmail.com]

Gerry got most of what would be my concerns about doing this in real life.  Each cut does remove wood too, that's why I was wondering if this was a thought experiment because in the original post you said to ignore it.

While an extra sheet or two of plywood does cost money, the extra time making all these cuts also has a cost in lost time.

I'm certainly not trying to discourage you.  These are challenges we all face.

The opposite direction of this project is the use of plywood in plydomes:  Finding a way to use full sheets with zero cuts.

Please keep us updated.

[Paul Kranz]

Normalson:

When I cut plywood to cover a triangular panel, I like backing off the plywood from the frame a little which is why I'm not concerned about the amount of sawdust. I agree that too many cuts would cause a problem, that is why I am trying to minimize the number of pieces.

My program scanned a 1/10-foot grid of a 4-by-8 rectangle and turned the intersection of the three 10-foot equilateral triangles for 360 degrees at each point and came up with the attachment. It is a good starting point, but it needs some work as you can see.

Paul sends...

--
--
You received this message because you are subscribed to the "Geodesic Help" Google Group
--
To unsubscribe from this group, send email to GeodesicHelp...@googlegroups.com
--
To post to this group, send email to geodes...@googlegroups.com
--
For more options, visit http://groups.google.com/group/geodesichelp?hl=en

---
You received this message because you are subscribed to the Google Groups "Geodesic Help Group" group.
To unsubscribe from this group and stop receiving emails from it, send an email to geodesichelp...@googlegroups.com.
For more options, visit https://groups.google.com/d/optout.

[norm...@gmail.com]

I like what that program did, that's a neat idea for a program by the way.

What is your method going to be for blocking under the cuts when making the triangles?

[Paul Kranz]

Normalson:

Thanks for the compliment.

I use 16-inch centers for the ribs, so I would just modify them under the seams. However, I didn't like that the suggested division causes a problem with 0.2055 sq. ft. of the plywood. I am going to keep working on it. There were over 1.5 million combinations to look at, but I think the one I started with was the best the program could do.

Pauil sends...

On Sun, Sep 7, 2014 at 10:44 AM, <norm...@gmail.com> wrote:

I like what that program did, that's a neat idea for a program by the way.

What is your method going to be for blocking under the cuts when making the triangles?

--
--
You received this message because you are subscribed to the "Geodesic Help" Google Group
--
To unsubscribe from this group, send email to GeodesicHelp...@googlegroups.com
--
To post to this group, send email to geodes...@googlegroups.com
--
For more options, visit http://groups.google.com/group/geodesichelp?hl=en

---
You received this message because you are subscribed to the Google Groups "Geodesic Help Group" group.
To unsubscribe from this group and stop receiving emails from it, send an email to geodesichelp...@googlegroups.com.
For more options, visit https://groups.google.com/d/optout.