Huh?   A Model of Space, Infinity and Flow

Working Draft  Copyright (c) 2005 - 2007 Jim Imboden

Chapter 2  The Model

I used to love mathematics for its own sake, and I still do, because it allows for no hypocrisy and no vagueness....

~Stendhal (Henri Beyle), The Life of Henri Brulard

The model I’m going to describe is based on one simple formula:  y = a^3/ (a^2 + x^2).  This formula has been around since the early 1600’s and has an interesting history of three prominent mathematicians of the 1600’s and 1700’s that studied it.  Chapter 13 is devoted to just the history of the formula, and while the history is fascinating it adds no value to the model.

The name of the curve that the formula develops is called “The Witch of Agnesi”.  In the research I’ve done I have found nine or ten different versions of formulas called The Witch, and while they all produce a similar curve, I am only using the one stated above.  This version of the formula was created by Maria Agnesi and has a drawing that shows her thought process.  This drawing begins the framework for the model.

Figure 2-1

I am going to start with her drawing and explain some of the basics. I will then begin to convert her drawing to a three dimensional model, step by step.

The math required is 7^th or 8^th grade with a small amount of trig thrown in.  I am including many drawings that I think will help in the visualization of what is happening.  Each drawing is a step towards understanding what is happening. 

If you are familiar with statistics you are probably thinking this curve looks like a “Normal” or “Gaussian” Curve”.  It was this shape that drew me to it, and while it looks like a Gaussian Curve it has its differences.  See chapter 12 for the differences.

This is a mechanism that would produce the original curve.

Figure 2-2

(C) is confined to move around the circumference of diameter a.

With this model I have three ways to draw the curve.

1.        At the top I can move [X] left (-) or right (+)

This changes angle (c) which in turn moves [Y] up or down.

This just like the formula states

2.        On the Left side I can move [Y] up and down

This also changes angle (c) and causes [X] to move left or right

3.        Turning the Knob changes angle (c) which causes [Y] and [X] to change.

A marking pen is attached to where point [X] and [Y] cross and will draw the curve

 on any movement of [X], [Y] or (c).

Figure 2-3  Click on Figure to see animation.

This shows the model with the curve drawn.

While I have not built this device I am certain it would not be hard to make and it would be capable of drawing the curve as shown.  Building the model would pose one problem that you may have already picked up on.  The horizontal lines (or bars in the model) and the line of bar for (c) would need to be very long just to produce the curve above.  To build it correctly would require lines or bars of infinite length. 

The fascinating thing about this model is if you could build it (with the infinite lengths) just turning the knob would give you a show beyond your wildest imagination.  If you were to start with the knob at the zero position (with ( c ) at the top) and start turning the knob slowly clockwise [X] will begin to move.  As it draws the curve you will notice that [X] starts moving faster to the right with each degree.  As you approach the 90 degree mark [X] would be miles away, millions of miles then billions of miles moving father and faster with each slight turn of the knob.  You are now at 89.999999999999999999 degrees moving faster and farther.  Then at 90 degrees you have finally reached infinity (whatever that is).  Amazing!  But what happens if I turn the knob .00000000000000000000001 degree further?  When you just begin to turn the knob past 90 degrees [X] instantly goes from plus infinity to minus infinity.

Making the Model Circular

The model we are going to build attempts to get around the infinity problem by making the model circular.  You might be thinking that this will really complicate the model and we will get bogged down in doing a lot of trigonometry, but actually the math is no more difficult than the original model.  The key to this is to work in radians instead of angles.

Some of the people that have seen this model struggle when dealing with radians.

If you are not familiar with radians or it has been a long time since you have used them, here is a quick refresher on them.  Note: “r” is the radius of the circle.

Figure 2-4

Working in radians allows me to talk in units of length that simply wrap around the model without working with angles and the trigonometry associated with them.

The use of radians becomes important in the next step of the model HUH 2; it is circular, not flat like HUH 1. 

Figure 2-5

In the 2^nd step model, HUH 2, we take the flat [X] plane that forces us to deal with infinity, into a circular world that hopefully will confine infinity but still retain infinity’s most important property of “length unknown and unknowable”.  HUH 2 takes a somewhat more rational approach, using “distance unknown and unknowable”.   We currently view space and infinity as pretty much the same thing.  When we reach the end of space we are at infinity.   If space were circular when we get to the end we simply start over with the distance traveled continuing to increase.  Fig. 2-5 shows what would happen if I turned the knob 90 degrees clockwise.  On the larger circle I would have made it ¼ the way around while on the small circle I would have made it ½ way around. 

Figure 2-6

Fig. 2-6 shows what a mechanical version might look like.

The circular form of the model does not draw the nice looking bell curve like the linear model does.  Actually the curve that gets drawn looks kind of ugly as shown in Figure 2-7.

Figure 2-7

The shape is developed from “Z” that is an offset amount from the base [X] and is trying to form the bell curve inside the starting diameter.

 

Figure 2-8 shows what the mechanical device would draw:

Figure 2-8

If you look at Fig. 2-6 and think about turning the knob, when you get close to the 90 degree point the model will begin to bind up.

Figure 2-9

Fig. 2-9 shows the beginning of the problem, as the knob turns from the center of the large circle from 0 to 90 degrees, pi/2 radians are needed.  But on the smaller circle pi/2 radians is 180 degrees.  When we turn the knob 90 degrees we get what is shown in Fig. 2-10

Figure 2-10

 I am going to have pi/2 too many radians and no place to put them when I turn the knob another 90 degrees.

We now have the core of our model.  This is a very simple visual model with the only problem being that it won’t work.

Summary:

1. Think in Radians.  If you don't understand them take the time now before continuing.

2. Think about how to deal with the excess radians (the next chapter has suggestions)

 

Chapter 3 will show what is needed to make it work.