Sanjoy Nath's Geometrifying Trigonometry Trying to Visualize Euler's Formula without using imaginary numbers - SanjoyNath/SanjoyNathGeometrifyingTrigonometry GitHub Wiki

We need to show picture first

We will show the Diverging and converging natures of the Expressions in Trigonometric powers through Autocad Programming of Sanjoy Nath's Geometrifying Trigonometry(C) definitions of + - x ÷ and also other well defined operations

Sanjoy Nath's Spiral(C) due to Cos(Θ) powers

Theorems on overlaps , Outlines , Inlines , merging of line segments play key roles on Sanjoy Nath's Geometrifying Trigonometry(C) which starts with the Analysis of pictorial thinking on different possible arrangements occur due to several combinatorial arrangements of positioning of line segments there. We start with minimum required line segments to reduce over burdening on CAD to analyse the scenarios of the pictorial arrangements .(We dont have super computers at hand) . While we see there are so large numbers of arrangements of line segments happening , there is the fundamental 4 symmetric orientations occur for every multiplications and divisions of Trigonometric Expressions simply Ratios (which are trying to define Rational numbers formulations) .In Sanjoy Naths Trigonometric Progressions(C) we will see that the sequence of pictures have some operator combinations which generates a trend of pictorial arrangements of line segments which have some kind of alignments and fits best with other considered conditions in power series. Since we are working with BIM on Steel Structures and Concrete Structures domain , we dont get sufficient time to analyse the mathematical properties hidden there. We are too much busy with the Geometric arrangements in Structural engineering and in the domains of Automated proper Drawings Generation where it is a big pain area to align several Engineering Symbols in right place among all of such possible arrangements there. Overlapping line segments means there are chancces of Text overlapping or welding symbols overlapping while doing Affine Transformations on the settings on paper spaces of CAD. While studying the positioning of these powers of Sin(Θ) , Cos(Θ) we have found that there are some periodic natures of spirals where some of these are diverging spirals and some of these are converging spirals for some given Theta Θ . We are working on some dynamic modeling of these periodic natures of spirals , overlaps , outlines , inlines in the process of Text positioning , line segment positioning while writing programs for Drawing corrections . Normally we are not researchers of mathematics but we are extensive users of positioning of Engineering Drawing Objects on 2D paper spaces. For Some values of SEED_ANGLE Θ we see that powers of Cos(Θ) generates Diverging outlines where PERPENDICULAR line segments are always visible and for every * operations (Any of the Sanjoy Nath's Geometrifying Trigonometry(C) variety of symmetries) HBHBHB...... HYPOTENUSE line segments overlaps on the BASE line segments so ultimately all the BASE LINE SEGMENTS get suppressed under the HYPOTENUSE line segments keeping the PERPENDICULAR line segments visible. OUTER SPIRAL forms in the process of powering of Cos(Θ) generates Diverging for Θ<45 degrees and the NARRAYS(C) forms due to powering of Cos(Θ) for Θ>45 degrees converges towards the SEED LOCKED SET(C) and the simple complementary behavior is found for powers of Sin(Θ) . With this thinking keeping in the mind we see that e^(iΘ)=Cos(Θ) + iSin(Θ) which is the EULER's Equation ultimately geometrized to the orientational geometry of symmetry on 2D paper space of CAD or Tekla Drawing screen. With this visual understanding on the combinatorial pictures of CAD geometries so formed through Sanjoy Nath's Geometrifying Trigonometry(C) algorithms we could see that for the powers of n on Sin(Θ) HYPOTENUSE Line Segments overlaps on the PERPENDICULAR lines so only the BASE line segments of FUNDAMENTAL LOCKED SET(C) are visible on top which shows the spirals. We call these as the SinVergence or CosVergences. Sometimes we call these as SinSpiral and CosSpiral. Some times we call these as SinTractions or SinSpansions , CosTractions , CosSpansions. These natures of spiraling due to different symmetric multiplications on the triangles (FUNDAMENTAL LOCKED SET(C)) we have not showing other 34 types of line segments yet since we dont have super computer at hand. Surely there are different summatuion conditions visible and different forms of EQUIPOSITIONING overlaps EQUALITY or parallel conditions can appear. We need to cross verify the consistency of Sanjoy Nath's Theory of Geometrifying Trigonometry(C) in light of existing Trigonometric expressions and also with consistency checking for all of these conditions in geometrically meaning ful ways https://en.wikipedia.org/wiki/List_of_trigonometric_identities#Double-angle,_triple-angle,_and_half-angle_formulae https://en.wikipedia.org/wiki/Chebyshev_polynomials
We started with this simple dream for the tesselations https://www.geogebra.org/m/Mr5gjXmR where https://en.wikipedia.org/wiki/Integration_using_Euler%27s_formula

some pictures to set here for verifications

    The Fractional Trigonometry: With Applications to Fractional Differential 

https://books.google.co.in/books?id=BftNDQAAQBAJ&pg=PA283&lpg=PA283&dq=some+wellknown+spirals&source=bl&ots=jgxkvvLY0y&sig=ACfU3U2P9vFKweSx-5XS2UnalgS1NTggIw&hl=en&sa=X&ved=2ahUKEwi97KiU8ojgAhUSfisKHVnhDCIQ6AEwE3oECAEQAQ#v=onepage&q=some%20wellknown%20spirals&f=false While Talking about the types of Symmetries we have found in light of Sanjoy Nath's Geometrifying Trigonometry(C) the mission to find proper alignments of Line segments on the 2D spaces on the Engineering Drawings to put missions started with the generation of rigorous geometric proof mechanisms on Geometric way for any given Trigonometric Expressions , we started with several fundamental understandings on these which are too much related to the Number Theorems. I dont have sufficient knowledges on these physical realities and the natures of current researches going on regarding spirals or Fractals but it is very interesting to study the Riemann , Euler , Deophantine that they have done so large number of calculations , plotted sufficient graphs through hand calculations on paper ithout using calculators. We are using Spreadsheets and CAD softwares to understand the natures of movements of line segments for every given conditions. With these facilities we could define the multiplication of numbers as ALIGN and SCALE TO FIT for two ratios. These Ratios work fine for Rational numbers so ALIGN AND SCALE TO FIT works fantastic for Rational numbers. For IRRATIONAL numbers we can force for ALIGNMENTS but we cannot guarantee for SCALE TO FIT. Some level of approximation will appear for the SCALE TO FIT cases so some slips will occur.With the analysis of infinitesimals we can get through the FITTING of line segments small even still that will work. From the chaotic pictures and the systems pictures i have seen in By Carl F. Lorenzo, Tom T. Hartley Fractional Trigonometry , i could understand that it is the Differential Equations report on the global aspects of actual four symmetries of Sanjoy Nath's Geometrifying Trigonometry(C) conditions. We want to examine these scenarios from the very basic levels of understandings geometrically which we can straight forward use in Steel Detailing and daily use mensuration.High level of Space Science is not going to help farmers or builders directly.We were trying to prepare noisy geometries through reasons. Reasoning for randomness is a long time challenges and if we can formally see the formal patterns of noises hidden inside the symmetries of nature simply due to ALIGN WITH SCALE TO FIT options can explain the smoothe or non smoothe curves of Zeta functions which are due to four different assymmetric mirror products instead of smoothe rotations happening there inside the systems. IMPORTANT IMPORTANT IMPORTANT https://www.mathpages.com/home/kmath620/kmath620.htm