In the previous post on phi, we looked at how the pentagon has strong associations to the golden ratio, given the high number of phi relationships found when a pentagon and pentangle are put together. Pentagonal symmetry is found everywhere in nature – many flowers have 5 petals, most mammals have a trunk with 5 major projections (4 limbs and 1 head), we have 5 fingers on each hand, 5 toes on each foot, and so forth. The Golden Ratio governs the construction of many objects in nature, and some might be wondering, having taken this all in, well: why? Why does nature choose the ratio 1.6180339 between parts of a whole, instead of other ratios such as √2, √3, e, and so forth?
Of course, it is simplistic to say “many flowers have 5 petals” – many also have a different number: irises and lillies have 3 petals, some delphiniums have 8; ragwort has 13; some asters have 21 whereas daisies can be found with 34, 55, or even 89 petals. Black-eyed Susans have 21 petals, and plantain has 34 petals. No golden mean ratio seems apparent in these varied examples, until we take a look at the following series:
Start a number sequence with zero, then 1:
Add these first two together, and write down the product as the next number in the sequence:
0, 1, 1
Now add the second and third numbers together and write the product as the next number:
0, 1, 1, 2
If you keep repeating this process over and over, the series produced looks like this:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987….
This series is known as the Fibonacci numbers. Let’s compare the ratio between one number and its successor in the series, like 8:13, which equals a ratio of 1.625. How about 89:144? The ratio is 1.617977. Now look at a pair from further down the series, 610:987, which equals a ratio of 1.618032787. It is apparent that the ratios between numbers seem to be settling down to a value very similar to phi. In fact, though this series never exactly produces phi, an impossibility since phi is irrational, the value becomes closer and closer to phi as you move on out down the line.
I remember upon learning that in the pattern of the human heart beat cardiogram, there is a short spike, where the ventricle closes, and a major spike when the heart muscle contracts — and the the distance between those two spikes in the cycle was phi! I walked outside my cabin into the darkness and looked up at the sky and said out loud “why?!” A little too much late night reading I guess.
It turns out that a lot of the preference in nature for the golden ratio comes down to optimal packing arrangement. Imagine a plant growing, and putting out a structure like a leaf. From the stem, there is a full circle around the plant, a 360˚ range, in which the leaf can be placed. Make this 360˚ rotation equal to 1.0 turns.
So, if the plant puts out a leaf, grows a little vertically, then puts out another directly above it, we could say that the plant is producing leaves at a rotation of 0 or 1.0. Obviously, this arrangement of leaves would not be so helpful since each leaf would lie directly atop the one below, thus blocking the sun effectively from reaching all but the top leaf. Also, the plant is only making use of a tiny fraction of the full 360˚ of space available to it for placing leaves. Let’s say then that the plant putting leaves out at 0/1 rotation wouldn’t be well adapted to most environments, and this arrangement of leaf placement would thus be selected against.
The next plant to evolve, say, puts out a leaf every half turn, a ratio of 0.5. Successive placements would form the sequence, 0, 0.5, 1.0, 1.5. etc. The plant would have two rows of leaves, and again, the leaves would be arranged in such a way as to preclude the sun from getting to most of the leaves stacked up below, giving leaves the most access to falling rain, or pollinating insects, as the case may be.
One can see that over time nature would tend to select for those plants that developed the most effective arrangement of leaves so that each leaf could maximally absorb sunlight, rainfall, and give access to pollinators. It turns out that the most effective ratio for packing leaves and other structures in as efficient a manner as possible is a rotation of 0.6180339….
This optimal packing arrangement strategy, giving the most ‘bang for the buck’ as it were, is present in many plant structures, from sunflower seed heads, to pine cones.
We’ll look more at how phi might be used to develop spiraling shapes in the next installment in this thread, part FIVE. For now though I want to take a break from the abstract and look at how I made use of pentagonal form in a couple of pieces that I made a few years back. Knowledge of phi, and the relationship of it to pentagons led to my attraction to the pentagon as a form, since it is essentially symbolic of phi in it’s arrangement.
When Client C first approached me to have some furniture made, the initial project was to be a replacement for an aging rattan table with a glass top she had in her dining room. It was a bit small and only seated three. I asked Client C about how large a table she wanted, and how many she might want to be able to seat, and after thinking about it for a few moments, said “five”. Well, hello Mr. Pentagon! This was just the answer I was hoping to hear!
First, as always, I start with drawing. I laid out the table in full scale on a piece of doorskin ply, and located the positions for each of the five legs using the same drawing technique I illustrated in part III of this thread. After some discussion with Client C of the various woods that might be suitable for this table, for the frame and legs went with Honduran Mahogany, a wood with some lovely working qualities that holds detail well for carving. After considering the possibilities, I decided that I liked the interplay between pentagon and pentagram, and decided to create a pentagram within the table. After some hours of thought about how to join the pieces together, I concluded that it would be best if the points of the pentagram came into the head of each leg as parallel to each other as possible. This led in turn to considering making the pentagram curvilinear.
I happened to have a couple of nice pieces of quarter-sawn Pacific Yew kicking around, a wood that would be an excellent choice for a bow due to it’s high elasticity and resilience, so it seemed like a good choice for a bow-like curvilinear part. I thought the orange-brown of the Yew would work with the Mahogany’s color. The five parts making up the pentagram of Yew, the way I approached it, was composed of glue-laminated sections. I sliced the Yew up into quite thin strips, less than 0.125″ (1/8″) thick, and glued them together. When glue laminating, a factor called ‘spring back’ needs to be considered – the more laminae, the less spring back one has, and the more pieces needing preparation. Less spring back is desirable since it makes for greater predictability in the outcome from the glue up. A balance was struck, I thought, by making each laminated Yew arch out of 5 pieces of Yew. I made up a bending form, and glued the pieces up using a type of glue -resorcinol -that did not allow for ‘creep’.
Coming from a timber framing background and thinking ‘structurally’ as always, led me to form the legs of the table much as I would do were I building a pentagonal cylinder. Not that i make pentagonal cylinders on any sort of regular basis! To increase the stability of the table, I flared the lower parts of the legs outward, along a parabolic curve. Treating the legs as if they were hip rafters, I ‘backed’ the outside faces of the legs so as to bring them into plane with the idealized pentagonal cylinder. The backing cuts themselves of course needed to vary in depth along the lower curves of the legs.
The skirts connecting the legs, made from 1″ thick Mahongany, needed a relief taken out from the lower edge to allow for someone seated to easily slide their knees and upper thighs underneath. I opted to profile the bottom of the skirt into an elliptical curve, blended with an ogee-like return at each side. The skirts were thus tallest in section where the greatest potential stresses would be, at the junction of skirt and leg. This was critical since the table had no lower tying stretchers. To attach the skirt to the legs, I used a sliding dovetail joint. The Yew pentagram pieces, interwoven amongst themselves in a series of half-lap joints like the lattice (kumiko) in a shoji, also terminated at the top of each leg. To make an effective joint I made these connect with a form of a wedged sliding dovetail. The wedges are just visible at the front edge of the dovetails. I used Wenge for the wedges, a good choice due to its 10-ton crush strength rating. The lower edges of the skirts along the ellipse and ogee received a beaded edge, similar to what you saw on the tansu doors in an earlier post.
The glass top allowed the joinery to be visible. I made up a template for the top in a pattern suggestive of 5 petals, and had the glass place make up a piece of 0.5″ plate glass to match the template. I couldn’t have the tiny points of each petal sticking out, so I made them re-entrant, each one marking the position of the leg. I originally wanted the connections between these re-entrant points to be connected by elliptical lines, however the glass place balked at that request, stating it was beyond their abilities or technology.
As it turned out, they made a small chip in one of the re-entrant corners when cutting out, and I wasn’t present for when the glass was delivered, so after I discovered it a day later, we had no recourse. Sure enough, 8 months later that little chip propagated into a crack running right across, and the glass top was replaced with a completely circular top. I was down in the US at the time and thus had no opportunity to intercede. I really liked the re-entrant corners and would have preferred to simply find a better glass place. The client is happy with the table nonetheless.
Just prior to making this table I had in fact been studying pentagonal layout in the form of a 5-legged splayed stool with through-tenoned stretchers. All the legs are backed so as to form a perfect pentagonal cone. While this construction might look simple in final assembly, appearances, as they say, can be deceptive. This was a lay out challenge, a step up in difficulty from Japanese 4-legged splayed construction, and took me about 3 weeks from start to finish, though not working full time on it. While typically the top might be made from a single piece of wood, and the legs through-tenoned into the top, I chose to go the frame-and-panel route for the top, using a Japanese wedged locking miter joint for the frame. In building, I often make use of this particular joint for such things as the hafu, or barge board joint, or in the tokonoma frame corner. There are several versions.
The bulk of the stool was made from Yellow Cedar, and the upper through-tenoned stretchers were Pacific Yew.
The locking joints took a while to make; they are the only good way I know of to provide a solid miter joint, and one not dependent upon glue for mechanical strength. The slender wedges are made from Purpleheart, and are both tapered and parallelogram in section. There’s a reason for doing them that way, and it’s likely that I’ll take that up in some future post.
The glossy sheen you can see on the surface of the cedar is the result of hand-planing. Yellow cedar is a wood which polishes quite well indeed. It is a crime in my view to sand this wood, or other cedars, since it muddies all the delicacy of the grain and ruins the wood’s natural chatoyancy, whether it is figured or not.
Here’s a picture of the joint apart, above at left, and another of the joint wedged up below left. I’m going to now let you know that this pentagonal stool had a curious, er, ‘outcome’. I initially tried to put it in a gallery, which, since they take 40%, meant I had to ask a pittance, or I had to price it so that it was all but unsellable. I took the latter route, rationalizing that it was good advertising, and sure enough, after sitting in the gallery for 3 months, it was unsold. I took it back and kept it in my shop. Then it came time to move a few months later and I had a garage sale. I was in an ebullient mood, and things were selling like hotcakes, much to my surprise. I had the stool sitting out there in the sale stuff, un-priced. I wasn’t sure if I wanted to sell it in fact. A woman picked it up and turning to me asked, “how much?”. For some strange reason I can’t really explain, other than being in a state of excessive good humor, I threw out a price of, are you ready for it?….
She opened her purse without a pause, gave me the $45 and was off. I’d didn’t think too much about it until a little later. I have no idea what has happened to the stool since then or how it is being used, but anytime I return to Gabriola Island, my fellow woodworking friends never fail to roll their eyes and say, “I still can’t believe you sold that stool for $45!!!” I just look sheepish.
Oh well. There are other fish to fry as they say. My next goal, in terms the study of splayed post construction, is the final frontier: 7-sided. I say this as heptagons are the only polygon of 10-sides or less that can’t be produced with the standard geometer’s tools of straightedge and compass, and thus are mysterious.
I’ve already done all the drawings and developed views, so it’s just a matter of time until I get around to it. Hopefully I can withstand the urge to sell this one for $45.