Thursday, October 15, 2009

Poly Gone?

This particular adventure started at the grocery store, when I bought a package of egg noodles. Not the most auspicious of beginnings I suppose, and it's not exactly a thrill-a-minute type of adventure, but what the heck, a start is a start. The journey has been good so far at least.

Later, when I went to make dinner, I tore the package of noodles open and what did I find inside, besides the dried noodles of course, but this little gem:

Not that a cardboard tray is especially valuable, but the shape - an octagonal hopper with unequal sides - now that was beautiful to me! I also realize there here lay a geometrical drawing challenge. So I set about drawing such a hopper, thinking that a wooden box in that form would look pretty sweet, and the faceted corners might even make for an object that would suffer less damage at the corners when knocked around. Here's what I came up with:

And a close up of the corner, exploded view:

While SketchUp can provide, by measuring directly off the drawing, the angles I need to make such a piece, simply to rely on SketchUp would be the wrong way to go in my opinion. A carpenter needs to be able to solve geometrical problems, on the job, with as few crutches as possible and with simple tools like a straightedge, compass, and framing square. I would say a pocket calculator would be a good inclusion as well. Still, having the model in 3D was handy because it enabled me to confirm that the results I could produce with 2D developed drawings for the various angular measures needed were correct.

With a through-tenoned hopper (or a finger-jointed one for that matter), there are 4 angles that need to be solved for:

1) face cut
2) edge cut (miter)
3) edge cut (butt miter)
4) top/bottom lines for mortise

With a regular plan, regular slope hopper, in which the corner in plan turns 90˚ and the boards, if plumb would meet at a 45˚ miter - the conventional hopper in other words - the kō-ko-gen method can be used to solve for all the angles fairly readily. For other polygons however, like this octagonal example I show above, the kō-ko-gen method does not apply. Too bad.

I had in the past constructed a pentagonal stool, the study of which had left me with all the tools to solve the developed drawing problems which apply to polygonal hoppers:

Still, when re-visiting a particular geometrical issue, in a new application, I like to refer to various books I have to see what methods they might offer, and as I mull over various ways of drawing the same thing, I can often come to new insights. Thinking that an octagonal hopper was a fairly ordinary enough form, it seemed likely that one of my western carpentry texts would deal with the layout issues for such a creature in some depth. Well, wrong.

I was very surprised as I looked through resource after resource and came upon almost nothing on this topic. Books on the steel square were pored through, Ellis's books on joinery, Walmsley's "Construction Geometry", Monckton's work from the mid 1800's. etc, etc.. Then I looked in my French and German layout books. And then I looked through the dozen or so Japanese layout books on my shelf. After all that searching, I had found very little material on polygonal hoppers. It wasn't as if they weren't covered at all - I found several books that had short sections on octagonal and hexagonal hoppers, but they only detailed the drawing methods to produce the face and miter joints. I was very surprised to see none of the books made any mention of the butt joint in this application, and, well, not so surprised perhaps that the 4th angle mentioned above, for tenoned or finger jointed boxes, was also not mentioned. Maybe they never existed and I am the first person to think of putting together a polygonal hopper with mortise and tenon joinery? Hah! I very much doubt it.

Now, as far as polygons go, it is not too uncommon to see hexagons and octagons, but other shapes are comparatively rare. Maybe all but non-existent. Further, once you get much beyond 6 sides in a splayed box, the butt-mitered and through-tenoned option begins to make less and less sense due to the angle of the butt joint becoming quite acute. Once you move beyond 10 sides, you are well on your way to coopering for that matter. So I can see why the most common cuts that might be employed would be the face cut and the miter cut, and I must conclude from all the materials that I have looked at so far, that anything otherwise was quite uncommon. It's a bit weird - a polygonal hopper does not seem like a particularly exotic proposition.

I drew another box to explore the topic a little further - a pentagonal hopper with through tenons:

A close up of the corner intersection reveals the appearance of a butt-jointed connection instead of a miter:

The corner of the board sticks up a bit from it's neighbor. I like the look actually - the circular array of points reminds me of a cutter in form, and I could imagine placing the hopper on a spindle and drilling enormous holes with it (I guess some carbide tips would help with that fantasy, and it probably wouldn't get far given the tapered shape, but 'oh well'):

The through-tenons on this box also incorporate a tongue and groove connection along the inside face, which, cleanly cut and accurately dimensioned, should make for a hopper which is watertight. Or, I could do a version of the pentagonal hopper, perhaps, which uses wedged sliding dovetails to seal the corners, as I did on this waterstone pond I made a few years back:

The edge on the bottom side of the pentagonal hopper I have drawn is mitered. I think it would be fun to make these two polygonal boxes, and I'll probably endeavor to do so soon enough. I'm calling the eight-sided one "Octabox".

At this point I have solved the 2D drawing issues to produce the necessary cut angles, and that is the main thing. If you can draw it, you can make it.

As for the more conventional hoppers, I did detail in four previous posts (titled "kō-ko-gen", parts I through IV) most of the geometry needed to solve for such beasts. I have however removed those posts from my blog and am currently compiling and re-editing that material, adding some new bits in, and forming it into a comprehensive article on regular plan, regular slope hoppers. This is amounting to 65~70 pages at this point, though I hope to condense it a bit. It will be thorough. I will make this material available for purchase soon enough, as part of a self-study series on Japanese layout I am putting together. I'm selling it because I have a lot of hours into the study and drawing work, and I should receive some compensation for that. The market can decide, as the pundits tell us.

This carpentry drawing series will be sold through a link I will put on the blog main page at some point soon. The self-study series will start with the hoppers, move to splayed-leg construction, and then on to roof study, with hip roofs and so forth. There will be an exam component for each module - those people who feel that they already have the material down say, for the hopper, can opt to not buy the study material and can simply challenge the exam. Passing the exam involves not simply pen and paper work, but completing the project in question, be it a hopper, or sawhorse, etc..

My hope, to be upfront about it, is that a series of study materials, like proverbial dangling carrots will draw more people into this rewarding area of carpentry material. Also, those that move along in their study of this material will, concomitantly, encourage me to forge on with completing further material for the comprehensive book on Japanese Carpentry drawing and joinery that i have been working on. And hey, I might make millions from such an endeavor and finally be able to afford that Bugatti I've been wanting. People would see me and would say, "oh hey, that guy's the hopper millionaire..." I'd need to hire bodyguards, and perhaps a helicopter would be a good idea. Yah, right!

Those readers who are interested in the first article, please contact me directly and I'll put you on the mailing list. I haven't set the prices yet, and they will vary with subject matter, but the comprehensive hopper essay will probably be about $20, and the exam around $5.00. I'll work out the details soon enough.

Thanks for dropping by today.


  1. " little gem" is the perfect words.... the octagonal hopper is a very familiar shape to me as it is the shape of a tapered bezel for emerald cut gemstone . i make them all the time in my work though i would say a tinny bit smaller than your noodle box. usually i make them in the range of about 15mm x 10mm by 8mm deep some times the parts get so small that the on the corner pieces the inside surface will be less than 1mm at the top tapering to nothing at the bottom .
    i can not bring any ideas to help from the jewellery field as i have to do it with out any gages as the parts are to small for any layout. does sketch up allow you to drag your angles of your taper and have all your parts auto change to the new angles? ( its called history in some programs) if so you may be able to figure out some % ratio for each angle increase or decrease. then you could make a chart. or you could really dive into the bottomless well and try to come up with your own trig/ko-ko-gen math formula .

  2. Chris:

    Being an avid user of Japanese Woodworking tools, thoroughly enjoy the blog. Definitely I am interesed in the hopper essay and the book whenever it is finished!

    Please let me know as soon as the first of them are ready for distribution.


    Paul Phelps

  3. Hi Chris,

    I've been unable to (easily) find your email address on this site... Could you point me in the right direction? I'm definitely interested in the articles you're putting together, and I look forward to catching your presentation in Brookline on November 7.

    Thanks for all the great work,

  4. gregore',

    I thank you again for your comments. I also was reminded of a faceted jewel by the octabox form.

    I'm not sure if you misunderstood my writing however, in terms of solving the layout for the polygon hoppers. I've already done that. I have no need to set up %-ratio charts or such with SketchUp. SketchUp was great for creating the form and playing with the configuration of the boxes. As I mentioned, while I can readily obtain the cut angles from the SketchUp drawing, I prefer to do so instead using 2D drawings that I can complete with the basic geometers equipment. i prefer not to become dependent upon a computer drawing program to solve carpentry problems, and it is a slippery slope.

    There is no 'kō-ko-gen formula' to figure out for these forms since kō-ko-gen is a method which applies only to 4-sided regular polygons. Since 95+% of carpentry and woodwork involves 4-sided construction which connect in plan at 90˚, it is a highly useful method to have in the arsenal. For other polygons, like the two I showed in the post above, which connect at other exterior angles (108˚ in the case of the pentagon, and 135˚ in the case of the octagon), other drawing techniques must be employed. kō-ko-gen does not work for other polygons. These polygon drawing techniques also work for 4-sided work, and in some cases involve similar drawing methods, however, all in all, kō-ko-gen is an easier method to remember so it is the one I choose for 4-sided work.

    And Paul,

    glad to hear you are enjoying the blog, and I'll put you first on the list for the initial carpentry drawing essay on hoppers. I appreciate your interest.


  5. Matt,

    thanks for pointing that out! I always see my e-mail address in the upper right of the screen when I visit the blog or write, so I assumed every reader could see it too. Wrong! I went into my profile section and discovered there was a box needing a tick-mark. All done -- just click on "View My complete Profile" under my mugshot, and the following page will have a link to my e-mail address.

    Now I'm wondering if others might have been trying to contact me and had a difficult time finding my address(?) I hope not. Again I appreciate you pointing that out Matt and I also look forward to meeting you at Brookline Library in November.


  6. Hey Chris,
    An intresting posts as always. I am happy to hear you are preparing the article about the hopper and I can't wait for it to become available. I am currently figuring out those angles needed for the hopper using only a sashigane and using the 'japanese triangle method' (There is probably a name for this system) to derive angles.
    Your efforts to share all this information in the way you do is highly appreciated. And I am happy to pay that modest fee for it. To return on my comment in my previous email.
    I wish you a lot of inspiration and clarity in your work for the comprehensive book you are writing.


  7. Hi Mathieu,

    nice to hear from you and thanks for the encouragement about the book - it's a big undertaking and not likely to make any best-seller lists, but i think it needs to be written. I'll put you on the mailing list for the hopper essay. I should have it ready by mid-November I estimate.

    By the way, the 'Japanese triangle method' with the sashigane does have a name - kō-ko-gen-hō (the rise-run-length method').



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