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butchering a polygon... what am I doing wrong?

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1.7K views 10 replies 6 participants last post by  jamsomito  
#1 · (Edited by Moderator)
I've created a MONSTER!

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I took the challenge upon myself to make some coasters, but as a regular 20-gon (icosagon, 20 equal sides) because my Incra Miter 1000HD was taunting me with it's quick reference guide:

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Turns out it's not so easy. I resawed and planed some stock to thickness and cut perfect squares to start. I made a bunch of hexagons to get warmed up - no big whoop.

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But then I tried the 20-sided polygon and I found my brain is missing the cells it needs to make this work. I have no idea what I'm doing wrong. I learned the quick-reference chart was indicating half-angles for if you're making a frame or something, so I'm actually using 18-degrees for my 20-gon. I even printed a template, cut it out, and used it to set up up my miter bar, stop, and blade.

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I just started cutting and it came out weird (actually 10 faces, but not equal length), so I tried again except this time I actually numbered the faces as I cut and got a worse result.

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Some strategies I've tried:
1. reference each of the 4-sides of the square, make associated cuts at 18 degrees. Reference new faces, do all 4 again, etc. Repeat 5 times and I should have a 20-gon. Nope.
2. reference one face of the square, make a cut, reference the newly cut face, repeat until 20 sides. Nope.
3. reference one face of the square, FLIP THE PIECE and reference the same face again, repeat for all 4 sides. Nope.

I'm completely baffled. I think it has something to do with my pointy-edges from starting with a square interfering with my stop. What's the technique here? Do I have to move my stop for every round of cuts? Surely there's a more mathematically proper way to do this to ensure 20 equal faces. I could just paste on the template and do it at my edge sander, but I want to try to get this right on the table saw. Any thoughts?
 

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#2 · (Edited by Moderator)
Lay it out with diameter lines for each pair of vertices. Draw the polygon connecting the vertices.

Put a sacrificial fence on your Incra and adjust so the gold won't get cut. Remove Shop Stop.

Cut the first face being sure to pass thru both vertices. Rotate to next face and cut to vertices.

Align vertices to edge of cut line on ZCI. This makes it easier to see where actual cut will go. Setting mitre gauge to 18° up instead of 18° down as you have it, will allow you to see reference marks and better align to ZCI slot.

After half sides are cut, set rip fence and rip remaining faces with saw set to finished face-to-face distance.
 
#3 ·
Thanks Mark, I see that approach could work. I think it still has several steps that rely on my eye - sure I could get it close, but I'm wondering if there's some kind of jig that would allow it to self-position for every cut?
 
#4 ·
Mak an 18° tapered stick. After the first side is cut place it so the cut ends where the tapered stick starts. Clamp. Cut, rotate & repeat keeping the last corner where the taper meets the fence.
 
#5 ·
hmm, thinking out loud….icosagon is stacked pentagons, not stacked squares.

Maybe try start with pentagon, and cut off the 5 corners, then use the new sides as reference for 5 more sides, and keep referencing the new edge until done?

Going to have problem with use 90° back stop as your reference distance, as need different length as you trim off corners. Instead need the back stop angled to catch each previous trimmed corner.

Another method would be using a jig the has fixed center point to ensure the radius stays same, and starting with circular blank.

Considered in brief moment, could be wrong …

Best Luck.
 
#7 ·
BTW, the 9° angle noted on your gauge is for when you are making a 20-sided frame. In other words, you cut both ends of each side of the frame at 9° which yields an 18° angle at each corner when joined together. You can figure out the angle for any number sides by dividing 360 by the number of sides and if making a frame, divide that by 2.

I simulated the cut using Sketchup just to see what it takes and how to align each cut. I started with a square that matches the final size desired. You basically start with one side of the square on the gauge, make the cut and rotate the piece so that the next cut uses the face of the previous cut and rotate around stepping from cut to cut. Once you have gone around once, you rotate to the next face that will result in a new cut. The challenge is that there is not a consistent place for a stop block to align each cut because sometimes there would be a corner hitting it and other times a face. The best option might be to draw a circle and then use a construction technique to mark out 20 points on the circle. You could also use sketchup to print a template that you could use to mark it on your boards. Instead of using the points to draw the sides, use these points as the midpoint of each side and align it to a mark on your miter gauge. Make sure that the first point is on one the edge you plan to use to make the first cut. If you carefully lay out the points, you may be able to use a stop block to set the position of the first cut and to mark the point on the gauge to align each center point.
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Note that once you get to the point where you have just a short side against the miter gauge, the cut gets a little dicey so you will want to have a good way to hold it in place as you push it through the cut. If you are going to doing a bunch of these, a sled with hold-downs would be a good idea.

Or you could buy a CNC machine. ;-)
 

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#8 · (Edited by Moderator)
So I did a little arts and crafts and it does indeed appear possible starting with a square and using a fixed stop block.

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The key is to reference all 4 existing faces first, then move on to the 4 new faces and so on. Using one original square face, then a freshly cut 18deg face will cause interference between the fence and waste part of the square and put the next cut inside the shape. Since a 20-gon has some faces parallel to and 90-degrees to eachother, there should never be any points against either the fence or the stop - only faces.

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Repeat all 4 sides 4 times and you have a 20-gon.

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Now, this is assuming all 4 original sides of the square become sides of the finished piece, so it has to be sized exactly. This is tough to get the stop positioned exactly right. I want to run this again with an undersized and oversized square to see how that affects anything.
 

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#10 · (Edited by Moderator)
Ah. I didn't try making cuts using the 4 original sides of the square first. If I am thinking about it right, the starting square has to be sized perfectly or the alignment block won't work. I would not want to make the several of those cuts with a miter gauge though, especially with the smallish 5" pieces you are talking about. This is a situation where I would cobble together a quick and dirty sled with a fence, stop block and hold down to keep it safer.
 
#11 ·
Nailed it! Mostly anyway.

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The length of the sides got a little off, I think because my initial squares weren't perfectly exact size. I found my cheap printer scales incorrectly in only one axis so my print was ever so slightly oblong, so since I used that to align my stop, could have been that too. Hmm, seems we still haven't found a way to completely remove the human element, but this is close enough for this project I suppose.

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I found it's best to rotate the piece counter-clockwise between cuts. This will give you the most surface area contact with your stop and fence for the longest time. Rotating clockwise, you rotate in one shorter face in cut #2/4 and have smaller faces for registration on both for cuts #3&4/4. Rotating counter-clockwise you have bigger faces for cuts #1 & #2, only one smaller face on the stop for cut #3, and finally smaller faces on both for cut #4. Minor detail, it does get tedious towards the last angle since all faces are small, but I had no problems with my miter bar setup. I hooked my left hand around over the stop and felt pretty safe this way.

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