Making an Octagonizer

Motivation

In more than one episode of “The Woodwright’s Shop” Roy uses a gauge he calls an “Octoganizer”.  See this recent show at about 23 minutes in. He can mark a piece of square stock with the layout lines needed to plane off the four corners, creating an octagon. The tool has a pair of locating posts that straddle the work piece, and two scratch pins to mark the face.

These screen shots from the Woodwright’s 3613 episode show the antique Octagonizer and also Roy marking a stool leg blank. He made a point that the tool can follow a tapered leg blank.

Bottom side of the Octagonizer

Bottom side of the Octagonizer

 

Using the Octagonizer

Using the Octagonizer

Searching the internet reveals this is a common tool in the boat building business called a “Spar Gauge”. I don’t know what “Spar” is on a boat.  I thought it was something Texans carried in the back of their pickup. Many internet pages discuss methods of making this tool, here is one that uses a graphical method to locate the marking pins.

I decided to explore the concept and make one. Or two. Or three. It turns out one size doesn’t fit all.

The Method

So exactly where do you drill for the scratch pins?

This is the necessary derivation:
In the following W = Width of stock, F = Width of a full facet, X = Width of an angled facet (to be removed).

Square stock layout

Square stock layout

The full width W contains one full sized facet and two angled facets
W = F + 2 * X

Angled facets measure full width times the cosine of 45 degrees, which is \frac{1}{\sqrt 2}
W = F + 2 * (F * \frac{1}{\sqrt 2})
W = F * (1 + \frac{2}{\sqrt 2})
W = F * (1 + \sqrt 2)

Rearrange the last to solve for the full facet width:
F = \frac {W}{1 + \sqrt 2}
Plugging in the numbers and calculating gives:
F = W * 0.4142

But we really need to know X, the width of the angled facet, so we can mark the stock by measuring from an edge.
X = F cos 45
X = \frac {W}{1 + \sqrt 2} * \frac {1}{\sqrt 2}
X = W * \frac {1}{\sqrt 2 + 2}
Running that through my calculator gives:
X = W * 0.2929

So 0.2929 is the Magic Number!

Just to verify:
0.2929 + 0.4142 + 0.2929 = 1
Yes!

Implementation

Locating posts on either side of the tool are a source of error because of their thickness. If the tool has to be skewed to a really steep angle, like using a four inch long Octagonizer to mark a half inch stick, the marks will be too close to the edge. In this exaggerated example with posts an inch in diameter, the scratch pins miss the thin board completely.

Error caused by peg diameter

Error caused by post diameter

If the locating posts were infinitely thin this would not happen and the tool could always lay out an accurate octagon. Therefore we need to keep posts as small a diameter as practical and avoid steep skew angles. I’m going to use six penny nails for posts and eventually make several Octagonizers to accommodate projects of different widths. Practically though, for many uses octagon shapes don’t have to be perfect.

The Octagonizer I made doubles up on a 4 1/2″ piece of Osage Orange. The wide side will mark stock up to 3 3/4″ wide. I let the wide side scratch pins stick out on the side opposite the points, these form the locating posts for marking narrower stock up to 1 3/8″.

Dual Octagonizer front

Dual Octagonizer front

 

Dual Octagonizer wide side

Dual Octagonizer wide side

 

Dual Octagonizer narrow side

Dual Octagonizer narrow side

 

This photo shows the wide side marking a piece of 2 inch stock. I’ve enhanced the scratch marks with pencil for the photo.

Marking a blank with the Octagonizer wide side

Marking a blank with the Octagonizer wide side

I had a piece of Poplar about 1 1/4″ square, I marked it out with the Octagonizer’s narrow side. Here it is clamped corner to corner in the vise.

Planing a 1 1/4" Poplar square into an octagon

Planing a 1 1/4″ Poplar square into an octagon

The Poplar works down quickly. I left one facet uncut just to show how it works.

The first try, three facets planed

The first try, three facets planed

While working through the arithmetic to locate the six holes in this double sided tool, I had to carefully account for the radius of the nails. Six penny nails measured 0.116″ in diameter, not accounting for this would throw the accuracy off a lot. I sharpened the points before assembly by chucking the cut off nails in a battery powered drill, then gently spinning them against a grinding wheel. The points were tempered by heating them red hot, quenching in water, then cooking in a toaster oven for 20 minutes at 425 degrees. I used a machinists vise to press the nails through pre-drilled holes in the Osage Orange.

Usage

In many cases you can set a marking gauge to Width times the Magic 0.2929, and just mark all eight lines with that.  If I had to make only one octagon I would use a marking gauge. If I had to make more than four, I might make an Octagonizer. A marking gauge will not encounter the error discussed in the previous section and you can lay out an octagon on a piece of stock any length, any width. It would not work though on tapered stock.

I plan to Octagonize a treated 4×4 for a porch support post.

Roy showed using the Octagonizer to lay out a tapered stool leg but laying out a short tapered octagon like a chisel handle, can also be done by marking both ends of a tapered blank, then using a straight edge to connect the dots. This is also a good method if you don’t want to see evidence of scratch marks.

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