Saturday, January 09, 2010

Torque and Seven Dimensions: What Does it Mean?

I have long been fascinated by the concept of torque or what the engineers appear to call moment. It never ceases to amaze me how torque increases with the distance from the fulcrum point. For example, a long screwdriver always produces two or three times the rotational force of a short screwdriver.

The physics are outlined in the Wiki:

But my question has always been: Why is this the case? Why should torque increase proportionally with a spatial extension from the fulcrum point? There seems to be some principle of dimension or basic math and geometry that I am missing. Clearly, the extension into greater space yields more force with application but again why that should be eludes me. What is the relationship between the extension in space of an object and the force multiplier when the object is rotated or lifted in some way? (Just to say that is the case or to insist “that is just how it works” doesn’t explain it for those of us who may be visually impaired, or in my case perhaps, mentally impaired.) Now it is more obvious with a crowbar but the principle is the same even though it is more apparent from the point of view of use. Force can be multiplied by distance. (It would seem that it is not related just to mass as you could have a short but very heavy crowbar, which might be the same weight as a long one and not get the same results.)

Gearing too works on the same principle but I feel like there is some utility here that remains unexploited and perhaps not as well-explained as it might be. For example, a flywheel operates on a similar principle and if hooked to the right gearing system can exponentially increase power from smaller efforts. (Note that the flywheel and a gearing system constitute a double principle of leverage.) None of this is anything new for engineers but by torquing the question, in this instance, with the long pole of big brains maybe the answer can be leveraged into a more pleasing and useful formulation. Why is it that the greater length of an object can produce more torque than a shorter one? Note that the physics below explains or delineates how this can be the case but the visuals don’t quite explain the operative principle. Please read the Wikis and don’t leap to conclusions about my intellectual opacity because they lead to something very interesting and dear to my heart as the developer of the metaphysical notion of dimensionally interactive cyber kinesis.

“ The magnitude of torque depends on three quantities: First, the force applied; second, the length of the lever arm[4] connecting the axis to the point of force application; and third, the angle between the two. In symbols:

τ is the torque vector and τ is the magnitude of the torque,
r is the displacement vector (a vector from the point from which torque is measured to the point where force is applied), and r is the length (or magnitude) of the lever arm vector,
F is the force vector, and F is the magnitude of the force,
× denotes the cross product,
θ is the angle between the force vector and the lever arm vector.
The length of the lever arm is particularly important; choosing this length appropriately lies behind the operation of levers, pulleys, gears, and most other simple machines involving a mechanical advantage.
Now if you look at the notion of a cross product this is where it gets really interesting. Here is the Wiki:

“In mathematics, the cross product is a binary operation on two vectors in a three-dimensional Euclidean space that results in another vector which is perpendicular to the plane containing the two input vectors. The algebra defined by the cross product is neither commutative nor associative. It contrasts with the dot product which produces a scalar result. In many engineering and physics problems, it is desirable to be able to construct a perpendicular vector from two existing vectors, and the cross product provides a means for doing so. The cross product is also useful as a measure of "perpendicularness"—the magnitude of the cross product of two vectors is equal to the product of their magnitudes if they are perpendicular and scales down to zero when they are parallel. The cross product is also known as the vector product, or Gibbs vector product.

The cross product is only defined in three or seven dimensions. Like the dot product, it depends on the metric of Euclidean space. Unlike the dot product, it also depends on the choice of orientation or "handedness". Certain features of the cross product can be generalized to other situations. For arbitrary choices of orientation, the cross product must be regarded not as a vector, but as a pseudovector. For arbitrary choices of metric, and in arbitrary dimensions, the cross product can be generalized by the exterior product of vectors, defining a two-form instead of a vector.”

Now stay with me. Here is the key that I may have been scrabbling after: seven dimensions.

“In mathematics, the seven-dimensional cross product is a binary operation on vectors in a seven-dimensional Euclidean space. It is a generalization of the ordinary three-dimensional cross product. The seven-dimensional cross product has the same relationship to the octonions as the three-dimensional cross product does to the quaternions. Nontrivial binary cross products exist only in 3 and 7 dimensions. There are no higher-dimensional analogs.”

What does this mean in layman’s terms? As a metaphysician, I feel there is something important here for reflection and further analysis but I don’t know enough math and physics to quite wrap my head around it. I feel like there is some “translation” of force I am not quite grasping.

Notice how even inquiring about such a question for collective input can become a kind of torque or moment. Collective intelligence appears to work in the same way—you get extension, you get a force multiplier but somebody has to ask the question to get the extension and then the attraction begins but that is probably another question.

Sean O'Reilly

President & Founder
Auriga Distribution Services

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