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Regular Polygons with Technic - How many can You build?

DrJB
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DrJB

DrJB

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One of my favorite topics is to build 2D/3D shapes using various connector types parts. I'm currently working on a project to reproduce the mechanics of a combine header/feeder. Such system uses a rotating pentagon to move the 'combs'. Now, Everyone on here knows how to built rotating squares and hexagons, but has anyone been able to build an equilateral pentagon?

I'm tempted to believe it is doable since, for many of us old-timers, we can draw a pentagon with just a compass and few pencil strokes.

Let's see how inventive/creative this community truly is.

 

 

technicmath

technicmath

Posted
Posted

In

https://webspace.science.uu.nl/~hooft101/lectures/meccano.pdf

by Gerard ’t Hooft the following is stated:

  1. "For a multigon of n edges, 10n-27 strips are required, using this method. With a bit more sophistication, I found heptagons made out of 35 pieces, 27 pieces, and any n-gon out of 7n-20 integral strips (assuming n to be odd; if n is even, a slightly different algorithm is needed). A bit of serendipity led to a heptagon of only 15 pieces (see Fig. 9)."
  2. "Figure 8: The rigid regular heptagon, using the method of equalizing angles as explained in Section 5. It is built out of 43 integral strips. This algorithm can be extended to any muligon."

 

astyanax

astyanax

Posted
Posted

That's a nice vid, but the solution with the T-shaped liftarm is slightly off. The T-shaped liftarm creates an angle of 106.26 degrees, not a regular pentagon's 108 degrees. :shrug_oh_well:

Also, it pains me to see all those cracked, rare & expensive red toggle joints! :cry_sad:

85x85p.jpg?1658325819.2117982

DrJB

DrJB

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  • Author
Posted
  On 11/30/2022 at 9:46 AM, technicmath said:

In

https://webspace.science.uu.nl/~hooft101/lectures/meccano.pdf

by Gerard ’t Hooft the following is stated:

  1. "For a multigon of n edges, 10n-27 strips are required, using this method. With a bit more sophistication, I found heptagons made out of 35 pieces, 27 pieces, and any n-gon out of 7n-20 integral strips (assuming n to be odd; if n is even, a slightly different algorithm is needed). A bit of serendipity led to a heptagon of only 15 pieces (see Fig. 9)."
  2. "Figure 8: The rigid regular heptagon, using the method of equalizing angles as explained in Section 5. It is built out of 43 integral strips. This algorithm can be extended to any muligon."

 

Expand  

Thank You. That's a beautiful reference. Just finished reading it, and there is also a Meccano Math 2, for mechanisms .... beautiful!

Lira_Bricks

Lira_Bricks

Posted
Posted
  On 11/30/2022 at 6:17 PM, DrJB said:

Thank You. That's a beautiful reference. Just finished reading it, and there is also a Meccano Math 2, for mechanisms .... beautiful!

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Just in case some people might not know where to look, there is a list-directory: https://webspace.science.uu.nl/~hooft101/lectures/

And there you can find https://webspace.science.uu.nl/~hooft101/lectures/meccano2.pdf and https://webspace.science.uu.nl/~hooft101/lectures/meccano3.pdf

aeh5040

aeh5040

Posted
Posted (edited)

There is a remarkable exact bracing of a regular nonagon:

IMG_20221202_124823321.jpg

and a 2-4-5 triangle has an angle within 0.2 degrees of that of a regular pentagon:

IMG_20221202_124734466.jpg

Edited by aeh5040
aeh5040

aeh5040

Posted
Posted
  On 12/3/2022 at 3:35 AM, DrJB said:

Thank you Alex ... I have none of the 7L thin arms ... lol ... a BL order I must do :)

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Of course you can use normal 7Ls with a +oo pin plus a 32054.png, but it does work out nicely with the thin ones.

astyanax

astyanax

Posted
Posted (edited)
  On 12/4/2022 at 6:28 PM, aeh5040 said:

Of course you can use normal 7Ls (...)

Expand  

Why not use normal 7L liftarms and normal black pins, then connect everything with 4L bars through the black pins...

Edited by astyanax
aeh5040

aeh5040

Posted
Posted (edited)
  On 12/4/2022 at 8:50 PM, astyanax said:

Why not use normal 7L liftarms and normal black pins, then connect everything with 4L bars through the black pins...

Expand  

Indeed, that's definitely also a possibility, although I always think of it as a bit inelegant!

Edited by aeh5040
DrJB

DrJB

Posted
  • Author
Posted
  On 11/30/2022 at 9:42 AM, astyanax said:

250x250p.jpg?1659341659.279528

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I thought this was an Efferman part (which I ordered only 2 weeks ago) ... In fact, it is a Lego part and came with the Airbus Helicopter. I've been away from this forum for so long and was not keeping track of the latest. Incidentally, such part is NOT available on the PAB website ... and commands hefty prices on BL.

msk6003

msk6003

Posted
Posted (edited)

Not for some kind of ~gons but can someone make isosceles right triangle?

No matter for size but smaller is better.

Edited by msk6003
howitzer

howitzer

Posted
Posted

Can't be done with Lego considering √2 is irrational number, except with approximations such as the suggestion above or maybe with axle joined with connectors and bushings as the hypotenuse.

aeh5040

aeh5040

Posted
Posted
  On 12/6/2022 at 8:53 AM, howitzer said:

Can't be done with Lego considering √2 is irrational number, except with approximations such as the suggestion above or maybe with axle joined with connectors and bushings as the hypotenuse.

Expand  

48250.png

aeh5040

aeh5040

Posted
Posted (edited)
  On 12/6/2022 at 1:02 PM, howitzer said:

That's not a triangle though.

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The middle section has two pin holes that are exactly 2 sqrt(2) apart. It can be used to make isosceles right triangles in many ways.

Edited by aeh5040
howitzer

howitzer

Posted
Posted
  On 12/6/2022 at 2:03 PM, aeh5040 said:

The middle section has two pin holes that are exactly 2 sqrt(2) apart. It can be used to make isosceles right triangles in many ways.

Expand  

But the elongated hole doesn't hold pins securely, instead they are free to move along it, so I don't think it's that great for building.

Davidz90

Davidz90

Posted
Posted
  On 12/6/2022 at 3:23 PM, howitzer said:

But the elongated hole doesn't hold pins securely, instead they are free to move along it, so I don't think it's that great for building.

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He meant the holes surrounding the elongated one. They are 2sqrt(2) apart.

howitzer

howitzer

Posted
Posted
  On 12/6/2022 at 4:48 PM, Davidz90 said:

He meant the holes surrounding the elongated one. They are 2sqrt(2) apart.

Expand  

Yes, so are some of the studs on any plate with at least the length of 2 in two directions, and holes in 3x5 beam and various frames, etc. But I'm not aware of any part that's correct length for the hypotenuse to complete the triangle.

aeh5040

aeh5040

Posted
Posted (edited)

The 2sqrt(2) length can be either the hypotenuse or the shorter side.  The other side is 2 or 4 respectively, achievable with a standard liftarm.

rt-isos11.png

Edited by aeh5040

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