Introduction into the Extra Geometry of the Three-Dimensional Space II

DOI: http://dx.doi.org/10.24018/ejers.2020.5.8.2016 Vol 5 | Issue 8 | August 2020 904  Abstract—Using the theory of exploded numbers by the axiom – systems of real numbers and Euclidean geometry, we introduce concept of extra plane of the three – dimensional space. The extra-planes are visible subsets of super– planes which are exploded Euclidean planes. We investigate the main properties of extra-planes. We prove more similar properties of Euclidean planes and extra–planes, but with respect to the parallelism there is an essential difference among them.


I. INTRODUCTION
We imagine our universe as the familiar three dimensional Euclidean space with its well-known apparatus of the ordered field (ℝ , ≤ , + , • ) of real numbers and the vector-algebra of ℝ 3 . Apparatus of exploded and compressed numbers (see [1], Chapter 2) is used, too, especially important is the concept of box-phenomenon. (See [1], Chapter 6. point 6.3 or [2], (8).) Other postulates, requirements, definitions, identities of explosion and compression are collected in the Part I of [2]. The idea of extra-line and extra-plane was alrady introduced in [1]. (See [1], Chapter 7, points 7.1 and 7.2).
An important characterization of the extra-line was mentioned earlier (See [3], Theorem 1.10.) Moreover [2] contains the characterization of extralines and the extracollinearity ( see Properties 1, 2 and 3) and the main results for extraparallelism of extra-lines. (See the Properties 4, 5 and 6.) The present article is a continuation of the foundation of extra geometry so, we continue Parts I, II and III with formulas (1) -(38), Fig. 1-3, Properties 1-6, Examples 1 and 1* and Definition 1 (concept of extra parallelism for extra-lines) to be found in the [2].
Similarly to the Parts I, II and III, we use the parameter = 1, so the explosion and compression is denoted without . The set 0 ; is called super-plane. (See (5) in Part I of [2]) The super-planes are situated in the Multiverse. Denoting that =̌ and using that =̌, =̌, =̌ , by (41) we can write 0 ; = { = ( , , ) ∈ ℝ 3 where 0 = ( 0 , 0 , 0 ) ∈ ℝ 3 is a given point and the (normal vector) = ( , , ) ∈ ℝ 3 with ‖ ‖ = 1. Clearly, 0 = ( 0 , 0 , 0 ) ∈ 0 ; . It is known that the equation , , , ∈ ℝ and = = = 0 is not allowed represents an Euclidean plane. So, the equation where , ℬ, , ∈ Ř = ℬ = = 0 is not allowed, represents a super-plane. (Super-operations were defined in Introduction into the Extra Geometry of the Three-Dimensional Space II I. Szalay and B. Szalay Part I. See Postulates of super-addition and supermultiplication.) If 0 ∈ ℝ 3 (⟺ 0 ∈ ℝ 3 , see (7) in Part I of [2]. ) then the joint part of the Euclidean plane represented by (6) and the closed cube (ℝ 3 ) may be a polygon, namely: hexagon, pentagon, square or triangle. In these cases, the super-plane represented by (7) is called as hexagonal-, pentagonal-, quadragonal-and triangonal super-plane, respectively. For example, if we have the Euclidean plane represented by the equation the joint part is a (regular) hexagon where 0 = ( 0 , 0 , 0 ) ∈ ℝ 3 is a given point and = (̌,̌,̌) ∈ ℝ 3 such that √ 2 + 2 + 2 = 1 As , are real numbers, using (2) in [2] with = 1 (see [2], Part I.) by (9) we can see, that the extra plane 0, , is represented by the equation where ( , , ) ∈ ℝ 3 such that √ 2 + 2 + 2 = 1 and = • 0 + • 0 + • 0 is a real number. Hence, the extra-plane 0, , is represented by the equation = tanh −1 (tanh + tanh ) ; −1 < tanh + tanh < 1and has the next graph which shows a hexagonal extraplane Of course, the border-curve of this hexagonal extraplane 0, , is invisible in ℝ 3 . On the other hand we are able to show it by piece by piece. For example, in the "depth" = (−1) and the "height" = 1 the holder of representing "level -curves". Their projections are seen in the "u,v" coordinateplane of ℝ 3. :    The joint part of the Euclidean plane and (ℝ 3 ) is a pentagon having the peakpoints , so the extra-plane 0, , is a pentagonal extra -plane   Hence, = tanh −1 (2 − tanh − tanh ) ; ( , ) ∈ ℝ 2 1 < tanh + tanh . Moreover, By (5) we can see that the superplane containing the extraplane 0, , has the equation 1 Hence, the border of the extraplane 0, , is a supertriangle, determined by the peak points So, the extraplane 0, , is a triangular extraplane, perceived by the following Fig. 6. Clearly, extra-complanar points are super-complanar points, too. The Eucledian geometry of Multiverse ℝ 3 partly covered in [1], Chapter 6, Sectiones 6.4 and 6.5. Here we have put together a more complete (i) all of them are in the universe ℝ 3 or (ii) one of them is in the universe ℝ 3 and the another two points are situated on its border, or (iii) two of them are in the universe ℝ 3 and the third point is situated on the border of ℝ 3 or (iv) all of them are situated on the border of the universe ℝ 3 such that these points do not situated on the same superplane of the border, then the boxphenomenon of the of the super -plane determined by the equation gives an unambiguously determined extraplane such that the superplane contains the points 1 , 2 3 . Proof. First, we remark that 1 , 2 and 3 are situated in ℝ 3 and they are noncollinear. (In the opposite case for the which by Axioms of Euclidean geometry is unambiguously determined and contains the points 1 , 2 and 3 . Third, we use that tha mapping between the points ( , , ) (̌,̌,) is simultaneosly unambiguous so, using the nominations =̌, =̌ and =̌ by (12) we have that the equation (11) determines a superplane which contains the points 1 , 2 3 . Finally, we observes that in the the cases (i) -(iv) this superplane and the our universe has a joint part, so, the points ( , , ) ∈ ℝ 3 satisfying the equation (12) form an extraplane. ∎ Remark 1. The reduced (11) is the best description of extraplanes by the equation  We may believe, that by the given points 1 , 2 3 the requested extraplane seems to be a triangular extraplane, but the Example 2 says that it is pentagonal.
Example 6. Let be given the point .
Using this transformation, the superplane has the equation Example 9. Let the extra-line be is the "w" axis of the rectangular Descartes coordinate-system and the extraplane given by the equation ̌⨁̌= 2⨀1 ⟺ tanh + tanh = 2 tanh 1 , ( , ) ∈ ℝ 2 , are second-type missing elements of our universe. It is sufficient to cast a glance at the Fig. 5.
The following example shows one of essential characteristic of the extra geometry.  These extra parallel extraplanes are shown on the next figure   .
The cutting extra planes We can see that