There is another way to view these layout lines as well, as I sit here pondering their meaning...

These lines can also be used as lines of balance, and as lines of balance they can also be stated as lines of strength. They are lines of weakness in a sense, and weakness can be harnessed to yield strength.

They are lines of balance not of each individual timber, but of the joint and of the building as a whole. The building is not any one of it's parts, nor is it merely the sum of it's parts. If we stacked an assortment of timbers on a pile, they would not create a strong building. Each one sits there unyielding, with no interaction with the other timbers all around it. Even if we stack the timbers in an orderly manner, we stilldo not have a structure.

The structure is a whole in and of itself. It's members combine to create something greater than their mathematical sum. They do this by sacrificing parts of themselves in order to strengthen the whole. This is what we call joinery; the art that we call ourselves the masters of. But if we want strength, then the wood should direct our cuts, not the line imposed by our will. We must be partners with the wood, not overlords. And this is why I believe that hand working is superior to machine work, it harmonizes and utilizes the strength of each timber, each fiber of wood. When we work the wood with our hands, then we can hear what it is trying to tell us. The master carpenter is he who has best learned the art of working harmoniously with the wood, of using the wood as his partner on the quest for perfection.

And this is the same with geometry. When we allow geometry to dictate our choices, then the structure will arise a harmonious, balanced unit. In balance there is strength, and there is harmony. I believe that the geometric design will prove the strongest in the end for this reason.

I have read that a hand hewn timber is superior in strength to an otherwise identical sawn timber, because the act of hewing utilizes the natural lines of weakness in the wood and as such the whole is left stronger, whereas the saw cuts blindly through and across the grain, ripping apart the fibers as it passes leaving gaping wounds. The cleanly sliced lines of a hewn timber are said to shed water, for example, and thus prolong the life of a cabin or exposed timbers in a frame built therewith, whereas the lines of a sawn timber trap moisture, and pull it deep within the timber. The saw yields weakness, which attracts fungus and insects, which yields further weakness...

So could it be too that a building laid out by the laws of nature is superior?

When working wood by hand, you exploit the lines of weakness. You split along the length of the grain where it is weakest, and as such you do not disrupt the natural flow of the wood, therefore leaving it stronger. I build bows, for example, and wen forming the back of a wooden bow I dare not cut across the grain of the wood in any tiny spot or else the entire bow will fail with a catastrophic explosion of fiber. Indeed, the smaller the error the more dramatic the failure might be. I must instead use the strength of the grain as my partner in this task, for only when the wood is at its strongest can it resist the tension placed upon it. And not only will it stretch and bend, but it will spring back with great force.

Perhaps geometry reflects this principle as well. Could it be that the lines of the daisy wheel are the lines of weakness? Could it be that by dividing your structure along its weakest lines, the whole is strengthened? Just as you divide the log along its weakest line to yield strength, so do you divide the frame along it's weakest lines.

It is said that true geometry -that is, the geometry of the circle- yields balance. The sum of the circle is zero, it has no beginning or end or limit, and the sum of the building yielded by it is balance.

And what of the axe? In my opinion the axe is the master of all tools. The axeman yields to the wood, all the while teaching the wood to yield to him. Just as the geometrician yields to the laws of nature, deriving strength from limitation.

Let's say that I am chopping a log in half along it's width. How should I do this? Does it not seem reasonable that I should stand atop the log and strike it with one solid blow with a good sharp axe, slicing it cleanly in two? We can all see that this will not work. Those who have used an axe know that the bit will travel a very very short distance before the resistance of the wood grain stops it. If we continue chopping like this, the log will eventually be cut but it will take a very long time. so how do we do this? We allow the log itself to become our partner in the process. We turn the axe at an angle, and sink the bit halfway with the grain, halfway against it. The wood itself forces us to conform to its patters, but all the while we are slowly making it conform to ours.

The axe is a tool of compromise, and compromise will yield a balance between strength and function. The strongest timber undoubtedly would be one split down its length, allowed to twist with the pattern of the grain or curve with the arch of the tree (because all trees have a natural arch, which at times as carpenters we should learn to exploit)or one that is left as a round timer, stripped of its weakness (bark and sapwood)-But these timbers would be difficult to join into a whole, it would be difficult to bring their strength into the strength of a structure. To yield the greatest strength as a whole, each member must sacrifice it's own strength. (this same principle, by the way, applies to us humans as well)

When we work with nature, then nature will be our partner along the way. -If we allow the grain to dictate our actions, then it will be our partner and our friend along the way, and soon we will learn to teach the wood to follow our path. That is the path of the axe, a tool of compromise yet at the same time a tool of strength and precision. We are again looking for the point of balance...

This all ties into geometry, or else I probably would not put it here at all. The lines of geometry are lines of balance, lines of harmony. And this all, ultimately, has to do with how we should apply these lines to reality.
If I am correct, then the geometric lines of a figure mark lines of weakness, and therefore they are lines of strength. As with the individual timber, we must exploit weakness to gain strength in the structure. By dividing a log between the week fibers of it's long grain, we yield 2 stronger halves than we would have if we had simply sawed straight through. Perhaps it is also true that if we divide the building along it's lines of weakness, then the whole will be made stronger. Perhaps there is more to this daisy wheel than mere aesthetic value. Perhaps there is more to these proportions than the fact that they are pleasing to the eye.

With geometry, we once again are striving for the point of perfect balance. We are allowing nature to be our guide and our partner once again, for what is geometry if not a reflection of the world around us?