It's pentagon time... awwwww yeah! The wild offspring of Mr. Square and Mrs. Hexagon; baby Pentagon is here to rock your world!

Lego thought: Don't you just love the sound of Lego bricks? I love the sound a pile of 1x2s makes when you reach in to grab the ones you want. I also love the sound of two bricks snapping together, there's just something so satisfying about that sound.

## Step 1: Parts

-90 1x2 bricks

Two color: 45 of each brick

Three color: 30 of each brick

Rainbow warrior: go wild

There should be a relatively simple relationship:

The brick-to-brick tolerance gap vs. the 1-stud side length tells you the angular tolerance, which determines both the extra opening or closing possible at each "right angle" corner, and also allows you to compute the incremental curvature along each side.

For the circle, the angular tolerance divided into 2.pi tells you the number of bricks needed around the circumference. Interestingly, your trial-and-error circles can tell you directly what the angular tolerance is -- divide 2.pi by the smallest stable circumference and that's the maximum angular opening.

For the polyhedra, the computation is more complex. There is a relationship between the straight (chord) length between vertices and the curved length you build, which can tell you the half-angle added onto the Euclidean polygon's corner angle. The sum of the Euclidean corner and the two half-angles must be close to 90 degrees, within the angular tolerance determined above.

Since the Euclidean corner is determined from the number of sides on the polygon [Q = pi - (2pi/N) = pi * (N-2)/N], the derivation above can be written generally for any regular polygon.

Once you've got that relationship -- angular excess/deficit from 90 degrees at each corner vs. curved length of side -- you can invert it and determine the minimum curved length possible given the angular tolerance.

It would be very interesting to know how close your constructions are to the limiting values computed as above.

For the angular tolerance, the 75-brick (1x2) circumference gives 4.8 degrees per joint. In that I'ble, you suggested that the 75-brick model might be close to minimal, so I'll assume 4.8 degrees is the maximum possible tolerance (over- or under-bend) on a 1-stud joint.

If you have successfully made undamaged circles smaller than that, the angular tolerance would scale up proportionately.

Your pentagons have 6.5-brick curved sides. Assuming the angular tolerance above, your pentagons should have interior angles of no more than 95 (90 + 4.8) degrees. A regular pentagon has interior angles of 108 degrees. That means the angle which each curved side makes with its chord should be 6.6 degrees = (108 - 94.8) / 2, and the total arc of the curved side is 13.2 degrees. With 6.5 bricks, the bend angle absorbed at each joint along the side is 13.2/6.5 = 2.03 deg, much less than the maximum tolerance; that's consistent with the picture, where the sides are curved much less than your circles.

The triangles have 7.5-brick curved sides. A similar analysis suggests that the curve-to-curve angle should be no more than 85.2 degrees (90 - 4.8). With interior angles of 60 degrees on an equilateral triangle, the chord angles are (85.2-60)/2 = 12.6 degrees, and the total arc of the curves is 25.2 degrees. Each joint along the arc absorbs 3.36 degrees, still less than the maximum tolerance.

Taking those analyses, I would predict that you could make your triangles with sides as short as 5.5 bricks (two bricks fewer per side), and your pentagon as short as 3.5 bricks (3 bricks fewer per side). In both cases, the sides would be curved close to their maximum (similar to the 75-brick circle).

For example, I show how to make a circle with 100 bricks per layer because it is easy to curve around and snap together (that's the black and white one shown). I make the lime green one using only 75 bricks per layer and that is harder to bring around and connect. Though I could go smaller than 75 per layer, I don't make the circles much tighter than that because I don't want them exploding in my face and the likelihood of cracking a brick becomes higher when they are under that much stress.

I also really like working with nice numbers (100, 75, 30, etc.) versus weird numbers (17, 23, 106, etc.) and it helps when making patterns to have multiples of 2 or 3.

As far as the limits go, I think there is a lot more play then a standard equation would show. Even when a tighter shape like the triangles are built, there is still a bit of room for things to bend and add uneven amounts of stress to different parts of the structure.