By studying in steps what a flat plane is, this paper shows that only six axioms are necessary for 2-dimensional Euclidean geometry until the Pythagorean Theorem: 1) the existence of stable space; 2) the existence of a straight line through any two points, 3) the existence of distance measurement between any two points; 4) the limitation of the space to 2-dimensional, 5) the repeated equivalence, and 6) the reflected equivalence.
I also use this to teach kids about math logic and math observation
This is the beginning of a worthwhile discussion.
Perhaps the followers of this thread would be interested in tackling variations of the Euclidean axioms from a physical space perspective. I suggest this because a view of the Euclidean axioms from a physical (tiny region, thick line) perspective rather than the abstract (point, line) perspective is more intuitive and more palatable.
For more about this physical space view of Euclidean geometry, see
Shape Descriptions and Classes of Shapes. A Proximal Physical Geometry Approach
Chapter Shape Descriptions and Classes of Shapes. A Proximal Physica...
Proximal Physical Geometry
Article Proximal Physical Geometry
and
TwoFormsOfProximalPhysicalGeometry20161608.06208v4
Data TwoFormsOfProximalPhysicalGeometry20161608.06208v4
The author intuitively, with many shortcomings, introduces the concept of flat space. He used the terms such as " repeat each other point by point", "stability of space", " the flat plane", " the flat surface", "moving", "rotation", "orientation", "translation", etc. without explanation and definition.
Using Axiom 2, we can define model: an incidence geometry for which every line has exactly two points.
Naturally, the following questions arise:
-Is the hyperbolic plane flat? The hyperbolic plane is homogeneous and isotropic.
-Why there exist at least three points that do not lie on a line?
-Axiom of completeness?
-Why the following statement is valid: If a line cuts one side of a triangle, and the extension of a second side, then the line cuts the third side?
-The terms "first rule" and "following rule" are unclear.
The axioms of order were implicitly being used in many proofs. This is one problem with "evident truths''; the author forgets to state some of the axioms and then the geometry is incomplete without them.
I am trying to clarify how did you prove the Parallel axiom (Theorem 34) based on the six axioms that he stated?
The paper is based on 59 titles from Wikipedia. Information from Wikipedia are not relevant and everybody can change them. Is there a published paper or book on which this paper is based?
Please refer to the Hilbert’s Foundations of Geometry.
Sincerely, Milan Zlatanovic
Sorry for my late reply. I was very busy lately.
This article is intended for my daughter, a middle schooler, so I tried to fit her understanding. Also, the starting point of each axiom or theorem is observation, which uses common language to describe.
Yes, I should have said: If two geometric objects have same set of points, they must be the same geometric object
The sentence " A same geometric object can be constructed repeatedly and unconditionally only if it is in a stable space." is descriptive for my daughter to understand, as part of the observation process. The strict part is Axiom 1, which implies that the space is stable.
"Flat plane" and "flat space" are all word describing observation. In fact, in the conclusion, it is clearly said that Euclidean geometry can not define flatness.
Other terms are used for describing observation, and they are not any part of axioms or theorems
Thank you very much for your response. I'm glad that your daughter likes geometry. I think that geometry in American high schools is not at the level that should be. Given that she is in high school, I think she can understand some parts of deductive approach, but it would certainly be good to approach the real world of Euclidean geometry in an appropriate way.
I recommend you a book "Geometry-Do" by Victor Aguilar. The book is methodologically uniform and the author systematically introduces, through problem solving, in axiomatic geometry.
The author presented all the material from a contemporary perspective, but also paid a long-term view of the classic point of view. In fact, he succeeded in achieving the ideal harmony of classical and contemporary exposure.
He gave a lot of application of geometric results in Architecture, specially the construction of gothic arches, then in civil engineering, Navigation, Military Science, etc.
I think it might be interesting for her and her friends.
If you have any question, do not hesitate to contact me.
Sincerely, Milan Zlatanovic
I have a new version of this paper with some fundamental changes. I wonder if it is now mathematically strict. Please have a look.
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