1.0 In Physics, a plane is the face of a solid
One thing the mathematicians never understood in the last 3,000 years is that there is no standalone
geometric figure called a plane. The word plane represents a category of figures, no different in this
sense than the word solid. The most popular planes are triangles, squares, and circles, all three of
which can be viewed as a series in an ever increasing number of sides: the more sides a plane has,
the more it resembles a perfectly rounded circle. Because of these misconceptions, the mathematicians
have developed the habit of rotating, translating, tilting, and scanning squares and triangles. The
scholars treat planes as standalone and tangible figures. These techniques may be okay for the
purposes of studying certain aspects of Mathematics such as limits and volumes. The problem is when
the mathematicians lose sight of what they're doing. The movement of a plane in any manner you can
think of is surrealistic and unrelated to Physics.
For the purposes of Physics, a plane is only the face of a 3D object! This is all that it means in Physics.
“ The 2D shapes are directly “inherited” from the 3D shape, as being constituted
by onedimensional sides fully belonging to twodimensional edges of the
original threedimensional object. In fact, 2D shapes are none other than 3D
shapes restricted to a certain perspective.” [1]
Go to the kindergarten room down the hall. Snatch a cylinder or a block from the kid's hand and inspect
it while he cries. You will note that the cylinder is capped by two circles whereas a cube is capped by
two squares. This is the only physical significance of planes in Science. A plane is naught by itself.
But now quickly tilt the cylinder or the square before the teacher arrives to find out why the child is
screaming. You will note that the end of the cylinder converted from a circle to an oval. If you tilt the
cube, the square magically converts into a rectangle. If you continue tilting the cube, at some point its
height or width should vanish and you end up with absolutely nothing: a 1D edge. This means that if
we attempt to tilt any solid sideways so ever slightly, perhaps in order to see the thickness of the face,
the plane invariably changes shape (Fig. 1). There is no way in Physics to tilt a cube from your
perspective and continue calling the face a square. A plane figure can ONLY be seen headon! As soon
as you tilt a square so much as a tad, you cannot continue calling the face you're staring at a square. In
order for the testimony of an observer to be objective, he has to tell us what he is staring at and not what
he did with the solid. When he is staring at the face of a tilted cube, he is NOT contemplating a square. If
we wish to use the words square and circle consistently in Science, we cannot change their shapes and
continue calling them squares and circles like relativists do routinely. A tilted square is either a rectangle
or an edge. It is not a square!
Okay everyone! Now in order to see how the tilting diet works, turn your heads like this. You should look 20 pounds thinner.

In Physics, planes do not stand alone and are merely faces of ideal solids. It is
irrational to say that you touched a square. The best you can do is touch the face
of what appears to be a perfect cube. When you tilt a cube, you are no longer
staring at a face you can call a square. You are now staring at a rectangle or an
edge. To continue calling the face a square implicitly makes your testimony
subjective. You are not stating objectively what you are staring at. You are asking
the jurors to imagine that you have tilted the cube in space and that they are to
continue calling the figure they're staring at a square as if they were looking at it
from an unspecified lateral location. This is unscientific because it delegates the
act of interpreting a figure to each juror.
2.0 Flatland revisited
Yet a much more popular habit the mathematicians have is moving planes around. They spin, swing,
and scan squares, triangles, and circles, usually to calculate volumes. Or they mold circles into squares
and back again, or cover spheres and cones with them.
In Science, however, if we so much as nick a square, it is no longer a square. In fact, you would have
trouble chipping or bending a square in the first place because it does not occupy space. A triangle is
an object because it has shape. On the other hand, a triangle can never be said to exist because it is not
allowed to have the property known as location. What has location is the pyramid the triangle belongs to.
The triangle is nothing more than one of the faces of a solid. Therefore, we can move a solid, because it
qualifies as an object. A plane cannot move from the solid it forms a part of. If ever a mathematician tells
you that a square escaped from his Rubik’s Cube, we know that it is time to call in the men with the white
jackets. Motion is a property restricted to objects that have location. Planes do not meet this requirement
because they are an intrinsic part of solids. They do not have location in and of themselves.
Therefore, the flat worlds described by Abbott, Ferris, and Heidmann are stealthily Solidland in disguise
and are therefore wholly misconceived.
Nevertheless, they should have realized a long time ago that Geometry is supposed to be a static
science. When a relativist talks about constructing a solid by spinning, translating, or swinging a
square, he is talking about the method to calculate a volume and not about what a geometric figure is:
plane: A category of rigid, continuous, smooth geometric figures which have in
common that they are twodimensional; A particular face or crosssection of an
ideal solid.
Now that we have a definition we can use consistently, let’s look at the two most famous planes.
Well yes, Bill! It's just a rectangle. But in Mathematics planes are malleable and dynamic.

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Copyright © by Nila Gaede 2008
You've got to be kidding me!

How do you tilt a square?
