It should be noted that, the following are attempting to depict two coordinate systems that occupy the same space. So, although BR ( Bottom Ruler ) and TR (Top Ruler ) are physically depicted as being apart, one must use their imagination to understand that this is only for identification purposes.
GIVEN a point A at 2, and two coincident Cartesian coordinate systems BR and TR.
Diagram A the active version
axiom 1
Coincident Cartesian coordinate systems possess equally aligned points.
It would be impossible to have Point A and no corresponding point in TR.
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Diagram B the passive version
No transformation has taken place yet.
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TR and BR analogy with a rhombic dodecahedron.
Given: two coincident Cartesian coordinate systems
Blue called BR having the Rhombic dodecahedron as shown in yellow.
Tangerine called TR, also having the Rhombic dodecahedron in yellow.
( The coincident tangerine TR system, with its' rhombic dodecahedron nor its' point B are not shown. )
photo 1
We place a coincident point A on BR and a corresponding Point B on TR.
If I shift ( therefore translate ) the tangerine Cartesian coordinate system TR over + 3 units, ( along the x direction ) then
1 The yellow rhombic dodecahedron still exists on the table in the blue BR
Cartesian coordinate system.
2 Point A on the yellow rhombic dodecahedron still exists in the blue BR
Cartesian coordinate system.
3 The yellow rhombic dodecahedron still exists in the tangerine TR
Cartesian coordinate system.
4 Point B on the yellow rhombic dodecahedron exists in the tangerine TR
Cartesian coordinate system
5 NO transformation has taken place yet, simply the translation of the one coordinate system TR.
The Galilean would apply the unmoved yellow rhombic dodecahedron in BR to make a transformation of an identical yellow rhombic doecahedron in that same location ( coincident location that shares equal points ) ... as the unmoved one.
The Galilean would ignore the existence of the translated rhombic dodecahedron in the tangerine TR Cartesian coordinate system
( with its' own tangerine rhombic dodecahedron ) as soon as the tangerine TR Cartesian coordinate system began translation.
While this transformation finds a value for x' given x, it should be finding out what x is in the other frame. However, by solving for x in TR, the Galilean unwittingly manufactures a new value for x' in TR. This value of x' [ aka point B ] in TR is ALREADY known, as seen in Diagram B. It is 2. It is 2 when the frames are coincincent. It is 2 when the frames are translated. It is 2 also, should the frames be returned to coincidence after the transformations.
By analogy, Point B no longer exists, since the rhombic dodecahedron in the tangerine TR Cartesian coordinate system no longer exists in their transformation scheme. Thus, prior to any transformation,
the "Diagram D before translation" scenario below, depicts the proper existence of Point B, prior to any transformation.
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Diagram C the Galilean results
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Diagram D before TR translation
axiom 2
Origin translation does not alter point location with respect to the Cartesian coordinate system in which it resides.
Point B still exists prior to any transformation.
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Diagram D after TR translation
Depicting both Point A and Point B requiring a transformation.
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Diagram E Galilean after translation and replaced to coincidence.
Since point A equals 2 and BR is now coincident with TR...then point B = 2 in TR.
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Diagram F Galilean after translation and replaced to coincidence.
( with Point B )
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Diagram G The Galilean's two possibles values or a point A transformed to TR.
Violates axiom 3. Point A transformed has either a value of -1 or 2.
axiom 3
A Cartesian coordinate system cannot possess equally named points having unequal coordinate values.
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Opinions, objections, questions, or comments regarding the above are invited. My address is swaterman@watermanpolyhedron.com
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