What is a circle?
Adapted for the Internet from:    Why God Doesn't Exist

    1.0   Euclid's twirling baton

    The circle has undergone significant modifications since the days of the Greeks without anyone noticing.
    Euclid defined a circle (Fig. 1) as a twirling baton:

    “ a plane figure bounded by a continuous line called its circumference or periphery,
       and having a certain point within it from which all straight lines are drawn to its
       circumference are equal.” (Bk. I, Def. 15)  [1]

    Euclid used the crucial word continuous in the mathematical sense: between any two points there is
    another one. Therefore, if his straight lines need to be drawn from the center to the circumference, they
    must necessarily end on discrete points and not on a physically continuous curved line. Euclid’s alleged
    circumference has no chance of being continuous. It is made of discrete beads one next to the other. The
    physical notion of continuity requires that the circumference be made of a single piece and not of discrete
    points. Therefore, his definition is an oxymoron.

    However, we have to concede that Euclid did call his contraption a figure and justified it by using the crucial
    word 'bounded' in his definition.


    2.0   The ring of apples AND the orbiting bead of Mathematical Physics

    The mathematicians have modified Euclid’s original definition significantly. A circle no longer consists of a
    bunch of lines that converge upon a center. In fact, it doesn't even consist of the center or of any part
    enclosed by the circumference. The circle of Mathematics can be either of two irreconcilable notions. The
    mathematicians use both notions in the same dissertation and get away with the best of all worlds.

    a. a ring of apples (Fig. 2)

    “ The set of points whose distance to a fixed point (the center) is a fixed positive
       number (the radius) is a circle.” [2]

    In order to know where to draw the next dot, you must plug in numbers into the computer, which
    calculates the location according to the equation:

                                                                            (x-x  )²+(y-y  )²=r²

    In other words, the computer calculates ONE location at a time, which means that you are allowed to
    place a single apple at a time. You are not permitted to see the circle of Mathematical Physics in a still
    image. The 'circle' of Mathematics is NOT a geometric 'figure.' The circle of Math is a series of apples
    that form a perimeter around absolutely nothing: The famous 'center' of the circle which is invoked to
    define the this geometric 'figure' is NOT a part of the circle! Only the equidistant 'points' are part of the
    alleged 'circle.'


    b. an apple that orbits around you (Fig. 3).

    “ A circle is the set of points in a plane that are equidistant from a given point O…
       The angle a circle subtends from its center is a full angle, equal to 360° or 2π
       radians.” [3]

    Recall that the magnitude of an angle is the 'degree of rotation.' This is the only notion of angle that
    has any practical use in Mathematics. The mathematicians have no use for the genuine, static angle
    of Physics (i.e. 'two lines extending'). Indeed, the mathematicians are so sloppy with their definitions
    and so inattentive to detail that sometimes they lose the apple altogether and continue to swing just
    the string around themselves. The Wolfram site depicts a twirling radius.

    But it only gets worse. Actually, it is not an apple swinging around a center, but an abstract location
    swinging around another abstract location called the center! The alleged 'point' that the
    mathematicians talk about is a dimensionless, static concept which the pictorial 'dot' is supposed to
    represent. Later, the mathematicians confuse the dot for the location, blend the dots into a physically
    continuous ring, and call it a circle.

    So? What should we conclude? Is a circle a bunch of lines extending from a center, a ring of apples lying
    still at the same distance from a center, or a single apple (or location) swinging around your head? Is a circle
    enclosed by a fence or by an itinerary? Is the enclosed area of the circle a part of it? Does the center point
    belong to the circle?

    In Mathematics anything goes. Anyone says anything and everyone else just nods. That’s how the idiots of
    Mathematics end up molding circles to the surface of spheres and saying that π is a variable.

Fig. 1
            
Euclid’s ‘twirling baton circle

(a photograph:  lines extending
from a center)    

Fig. 2

The circle relativists
advertised        
(a photograph:  points
equidistant from a center)

Fig. 3

The relativistic circle that you got in the mail    

(a movie: a single apple swinging around an apple located in the center)

    3.0   The circle of Physics

    In Physics, it is not even half as complicated as all of this. In Physics, we don’t define circles or triangles,
    or, for that matter, any geometric figure. We point to a circle and say the word circle (Fig. 4). The ET now
    visualizes what we are referring to and identifies the word with the indicated image for the rest of the
    presentation. End of story!

Fig. 4   Circle

    Now, whether the mathematicians can use this object or not is irrelevant. This is what a circle is! This is a
    geometric figure!


    4.0   Conclusions

    To summarize, for the purposes of Physics, a plane is a category of objects. More specifically, it includes
    the ‘set’ of two-dimensional objects (e.g., triangles, squares, circles, pentagons, parallelograms, ellipses,
    ovals, etc.). These figures:

    1.       do not stand alone, but are the faces of three-dimensional objects: the solids
      
    2.       are not infinite in extension, but rather have boundaries

    3.       are not made of points or lines, but are rather continuous and smooth – they are
    made of a single piece

    But here’s the punch line, and you will hate me for this. I have analyzed points, lines, and planes for the
    last few pages only to tell you that they are for the most part irrelevant in Physics. We don’t need any of
    these categories to explain any important phenomenon of nature. These worthless figures are needed
    only in the religion of Mathematical Physics. In Physics, we deal solely with solids.
But aren't you guys just
going around in circles?
Yeah, Bill! You're missing
the point! This is how we
make circles in
Mathematical Physics.

o
o

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        Copyright © by Nila Gaede 2008