Introduction into the Extra Geometry of the Three–Dimensional Space I

Abstract

Using the theory of exploded numbers by the axiom–systems of real numbers and euclidean geometry, we introduce a geometry in the three–dimensional space which is different from the euclidean-, Bolyai – Lobachevsky- and spherical geometries. In this part the concept of extra-line and extra parallelism are detailed.

Introduction into Extra Geometry of the Three-Dimensional Space II

“…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…”

“…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].…”

“…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].…”

“…(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].…”

“…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.)…”