GeometrifyingTrigonometry

We Are Doing Geometry and Trigonometry from school days but we were doing Trigonometrifying Geometry which means we were converting Geometric problems into Trigonometric problems to solve for some values to check these are giving some value of length or values of Area or Values of angle . Trigonometry solves Geometric Problems by converting Geometry into accessible or equatable algebra.In this same way we were proving several identities of Trigonometry which is also to compare some length values(As scalars) or Area Values(As Scalars) Or Angle Values(As Scalars)

BUT BUT BUT

if we want to retain the geometry intact and proceed with geometry and if we want to visualize all possible orientations that occur due to Trigonometric expressions or identities , That is if we want to do some Step wise proof of Trigonometric Expressions in terms of Geometry itself then we are stuck This Framework is a small endeavour to achieve this since it ie necessary for many 3D calculations and space planning in CAD or CAD related Softwares like BIM

Follow WIKI of this Geometrifying Trigonometry for more details https://github.com/SanjoyNath/GeometrifyingTrigonometry/wiki

https://geometrifyingtrigonometry.quora.com/ https://geometrifyingtrigonometry.quora.com/Geometrifying-Trigonometry Refer

GEOMETRIFYING TRIGONOMETRY IS ALGEBRA OVER LOCKED SET(FOR TRIGONOMETRY 2D IT IS HEP ARRANGEMENT AS DEFINED) (https://geometrifyingtrigonometry.quora.com/Algebraic-Structure-of-Geometrifying-Trigonometry)

Algebraic Structure of Geometrifying Trigonometry Before starting the Precised Definition of Geometrifying Trigonometry

While working on the more precised definition of Pivoting action of * operation for Geometrifying Trigonometry we have seen that ∞ (Pivoting with Aligning With Scale To Fit ) done between second HEP Arrangement(Locked Set) on superimposing with First HEP arrangement(Locked Set) need some more specific understanding. We have seen that there are 7 possible configurations can occur due to multiple usages of / and \ operation on second HEP arrangement(Locked Set) Is Sin(Θ) x Cos(Θ) equal Cos(Θ) x Sin(Θ) when interpreted or drawn geometrically?

Every multiplication in Trigonometric expression convertsinto * operation when it comes to Geometrifying Trigonometry Domain. Geometrifying Trigonometry precisely says we can replace * operation with any of these 7 operations ∞ , ∞/ , ∞/\ , ∞/\ , ∞//,∞,∞/ and ∞/\ after writing several conditions through Autocad programming Simulations we have found that Sanjoy Nath(c) Conjecture that ∞ is independent , ∞/ has same effect as ∞/\ , ∞\ has same effect as ∞// And ∞/\ has same effect as ∞/ So ultimately if we categorize the output configurations then we see there are 4 types of unique symmetric cases generated due to 7 different types of orientations or flips this gives insight that we can proceed with only these four conditions of * operation. If we use Regular Expressions for Syntax chancking on the Geometrifying Trigonometric Expressions , then we see there are Kellne Star kind of options occur on / and \ That means even we increase numbers of / or \ operations after ∞ , we are not going to get any new configurations other than these given four configurations

After doing extensive geometric simulations through Autocad programming , we have seen that Locked sets are defined as block of line segments where we will never scale these blocks with different proportions.If we have to scale these blocks , then we will do the scale_x=scale_y=Scale_z every time and we will never explode these locked sets

We have seen that if Trigonometric expressions have + before that then doing ∞/ (Same as ∞/) gives better result and if - is there before the trigonometric expression then ∞\ (Same as ∞//) gives better results.

We preffer ∞/ or ∞\ operations normally due to above reasons since these will give us unique outputs and unique inverses in closures of Line Segment space

Now natural Question arise is "What is the definition of best result for * operations?" We clarify this with the concept that Suppose we have to do HP*HP then the better result of * is that which gives output line segment exactly overlaps on the line segment H (Since H is the first line segment of Geometrifying Trigonometry Duplets)

For every computable resolutions of any topic , we need to provide a good well defined Structure

GT = { L , σ , D , Σ , * , . , +,-,x ,÷ , = , µ ,■ ,τ}

L ={Set of Locked Arrangement Sets}

Σ={Set of Line Segments free line segments or Line Segments well defined and lockedly arranged conformally inside the well defined Locked Sets}

D =Doublets sets defined in the Given Locked Sets{Locked set of line segments) . Each Doublet represent the trigonometric ratios in terms of (name of denominator) .( name of numerator) forms Example Sin(Θ) => H.P , Cos(Θ) = > H.B , tan(Θ) => B.P etc... If we need additional line segments σ , then we define these accordingly for given problems

. is the Operator which works inside given Locked Set and it cannot work outside any given Locked Set Arrangement since it is the mapping of Denominator Line Segment to Numerator Line Segment in the Trigonometric Ratios

L is set of Locked Sets used in the given Trigonometric Expressions . To represent the whole Trigonometric Expression (Given) in Geometrifying Trigonometry form we need to identify and understand number of Different Angles (Example Θ , δ,Φ,α...) are present in the Trigonometric Expression.

If somehow we can know the number of different angles involved in the Trigonometric Expression , then we can say the numbers of Locked Set Arrangement necessary to formulate the GT String(Geometrifying Trigonometry Stringology representations)

Next comes the question of numbers of Doublets involved. If in a given Locked Set Arrangement , there are n Line Segments pre arranged and locked then we need (nC2) x (2!) numbers of Doublets .Reciprocal of Trigonometric Ratios are defined as Reversed Doublets . Every Alphabets in the String Doublets represent a pre arranged and locked line segment in Definitions of Geometrifying Trigonometry

DRAG of HEP Locked Arrangement [we are working to define this more precisely]

µ (alt 230) MOVE or DRAG (Representing FREE MOVEMENT OR FREE DRAG operation on whole Locked Set independent of * operations)[NORMALLY WE DONT ALLOW FREE DRAG , FREE ROTATION OF HEP ARRANGEMENTS]

τ (ALT 231) TORQUE or ROTATE (Free Rotation of the whole Locked Set independent of * operation) [We will define these more precisely here after some more workouts]

■ (alt 254) FREE SCALING IS DEFINED ON FREE LOCKED SET AS SCALAR ALGEBRAIC MULTIPLICATION ON WHOLE TRIGONOMETRIC EXPRESSION THAT DEFINES SOME LINE SEGMENTS mapping.in -&nbspThis website is for sale! -&nbspMapping Resources and Information. THIS CASE WHOLE LOCKED SET(ARRANGEMENT) SCALES PROPORTIONATELY ALONG ALL DIMENSIONS

= Equal to Sign in Our normal Daily use Trigonometry simply evaluates and compares Length value (Scalar value) of two calculated data for given angles BUT in reality when we try to think the fact geometrically , then we see that Geometry can compare things with pure ruler and compass.If we think in this way of comparizations then we see that EQUALITY is not simple comparing of lengths values as scalar things

We see that STRONG EXACT EQUALITY means the line segments are exactly overlapping on one another where as GOOD EQUALITY is defined as the line segments are COLLINIER but not overlapped and equal in length , GOOD Equality is Parallel and equal in length , ORDINARY EQUALITY means the broken lines not on single collinier line are summed to make equal length where sum is allowed only when these LINE segments are not parallel.HERE Geometrifying Trigonometry is STRICTLY DIFFERENT from vector algebra because

in case of GEOMETRIFYING TRIGONOMETRY we cannot sum or substract the Line Segments if they are not parallel .Sum means + which is start-end - start - end chain of the line segment(This is like Toe - Tip - To - Tip chain... of Vector sum) and the substraction is also possible if the line segments are parallel to each other. If Two line Segments are parallel to each other then we can Translate the start point of one line segment on the end point of another parallel line segment (DRAG operation which we are defining more precisely in some other theorems)

We have checked that many of such operations are not completely cross verified yet in CAD programming.We are working with checking of Associativity for all 7 types of * operations and also for Other operations.We are trying to search best algorithms or redefinitions regarding PIVOT , ALIGN,SCALE TO FIT of ∞ operation where as we have seen through simulations that ∞/ and ∞\ works best

[we are working to define this more precisely] this is refined when we continue to write programs for 3d softwares.We are defining here with more refinements because we are not only mathematics lovers but we do it for practical purpose.We have practical purpose at hand directly used for clients needs.

in GEOMETRIFYING TRIGONOMETRY LOCKED SET IS CONSIDERED AS GEOMETRICALLY TRIGONOMETRICALLY ARRANGED PREDEEFINED LOCKED SET WHICH RETAINS PROPORTIONS AND ANGLES BETWEEN EACH OTHER ON ANY KIND OF OPERATIONS DONE ON THAT

x (Simple x symbol) and ÷ (ALT 246) are the scaler multiplications on LOCKED SET . The multiplication with scalar are not allowed directly on Free Line Segments in Geometrifying Trigonometry . When we scale up or scale down , we actually do that on any Locked Set (GEOMETRICALLY TRIGONOMETRICALLY ARRANGED PREDEEFINED LOCKED SET is SIMPLE TO UNDERSTAND as AUTOCADs BLOCK object having some predefined LINE SEGMENTS PROPERLY ARRANGED THERE)

scalar multiplication (x operation is nothing but doing Scale /scaling of Locked set) or scalar division (This is also multiplication with reciprocal of the value ÷ ) So obviously we cannot multiply or divide any scalar number with LOCKED SET in GEOMETRIFYING TRIGONOMETRY . SO to make the definitions precise and simpler we consider that all possible types of x and ÷ are done on right side of trigonometric expressions

While writing programs for Geometrifying Trigonometry (As we are writing for Autocad, For Tekla , For Revit , For Dynamo , For Grasshopper,For Rhino ,For Octave ,For MatLAB,For Blender...) we will push all left side scalar type number objects to right side of Trigonometric sub expressions.Sub expressions are the Trigonometric strings seperated with + or - symbols

Additions have 8 types of Arrangements and also - Substractions have 8 types of arrangements. We cannot Add nor Substract any scalar number object in Geometrifying Trigonometry.(+) Additions or ( - ) Substraction always work on some LOCKED SET when particular LINE SEGMENT OBJECT INSIDE that locked set is PARALLEL OR COLLINIER to OTHER TRIGONOMETRIC SUBEXPRESSION s LINE SEGMENT Actually Additions or Substractions can work on line segments but line segments cannot move freely on space (As per GEOMETRIFYING TRIGONOMETRY) , so obviously any line segment can travel anywhere in 2D space (We will extend this to nD on some next time) AS AN INTEGRATED LOCKED PART OF ITS LOCKED SET (Think it As Autocads Block object )

TOO MUCH PRECISE DEFINITION OF ∞ (ALT 236) OPERATOR We know we have already defined 37 common line segments used in Trigonometric Expressions from several exercise books 37 Line Segments and 10 Points Identified to Symbol set of Geometrifying Trigonometric Formal System

We know ∞ does three things on second HEP LOCKED SET OBJECT to Superimpose one line segment of second HEP Arrangement(LOCKED SET) on the other line segment of First HEP Arrangement (LOCKED SET) where THE MOST CRITICAL OPERATION IS ROTATION

ALIGNMENT of one object with other is simply PIVOTING + ROTATING + SCALE TO FIT , where we consider ROTATION as first task that ∞ does.MINIMUM ENERGY ROTATION is the key factor here for ALIGNMENT . We choose pivoting later but we choose the direction of rotation first which will align the line segment of second HEP (Locked set) to overlap on the line segment on First HEP (Locked Set)

Suppose as per definition of Geometrifying Trigonometry Stringology Conventions , we convert the whole Trigonometric expressions into STRINGOLOGY forms examp,e (Sin(Θ))^2 [Read Sin Squared Theta as Sin(Θ) x Sin (Θ ) as usual notations] is turned into H.P_ΘH.P_Θ (here we have intoduced _ (Underscore operator to identify the GOVERNING ANGLE OF PARTICULAR LOCKED SET ARRANGEMENT), suppose a given trigonometric expression is with only one angle (GOVERNING ANGLE say Θ) considered , then the GEOMETRIFYING TRIGONOMETRY STRING simplifies to H.PH.P or more simply HP*HP

here in the expression we see that HPHP has a substring of PH sub part . With this we simply understand that the multiplication in Trigonometric expression is turned into * operation in GEOMETRIFYING TRIGONOMETRY which has 7 types of varieties as discussed earlier but we are concerned with ∞/ or ∞\ only for normal school level geometry cases.For proffessional programming cases we consider the other types of operation also NOW LETS FOCUS ONLY ON ∞

never confuse this ROTATION with τ (Free rotation of LOCKED SET ARRANGEMENT OPERATION)

This ROTATION is inharent inside ∞ operation which has two types of cases possible one is MINIMUM ENERGY ROTATION and other is MAXIMUM ENERGY ROTATION . WE HAVE USED THE MINIMUM ENEERGY ROTATIONS for ∞ operator which means more angle covered in rotations means MORE ENERGY USED to rotate and if we use less angular travel to align line segments , then there is LESS ENERGY USED . In this terms we have planned the rotation to do LEAST ENERGY SPENDING of angular rotation of one LOCKED SET to superimpose LINE SEGMENT on ANOTHER

Example of [Sin(Θ)]^2 means H.PH.P where PH is there means Right Side line segment H has to rotate (90-Θ) degrees to become vertical (Aligned on Perpendicular) which is anti clock rotation.If we choose clock wise rotation , then We have to Cover angular movement of (90+Θ) degrees which means For Sin(Θ) Case anticlockwise rotation gives LESS ENERGY ROTATION so we will rotate the Second HEP Arrangement (LOCKED SET Simplest form of LOCKED SET HEP ARRANGEMENT IS A RIGHT ANGLED TRIANGLE Having H , P, B Line Segments only)

From this example it is clear now that we are chosing Rotation operation intrinsic inside ∞ operation with specific calculations which we have put into our AFFINE TRANSFORMATIONS MECHANISM OF ALGORITHM FOR GEOMETRIFYING TRIGONOMETRY) then we do scale to fit Hypotenuse of second (Right Hand Side GT String Locked Set) Locked Set on the Perpendicular of Left Hand Side Locked Set Since PH substring is visible in the string [[[ RECALL H.PH.P where P*H is explained in detailin last line here]]] Our algorithm searches the Pivot points accordingly [check hyperlink above]

Now we want to explain the scalar multiplications in GT STRINGS .Here we have designed our scalar multiplication as BODMAS typr of OPERATOR PRECEDING MECHANISM .Strict working rule that we have followed here is that we will do all kind of scaling ( scalar x or ÷ from right side that is When all geometric operations and Affine transformations on Trigonometric expressions * operations like[whichever is best or all possible actions ∞ , ∞/ ,∞/\ , ... done] NEVER DO ANY SCALING OPERATION UNTIL THESE ACTIVITIES ARE DONE

SIMILARLY , NEVER DO ANY + , - ACTIVITY UNTIL ALL SUB EXPRESSIONS (WITH BRACKETS , *) ARE DONE . WE HAVE TO KEEP IN OUR MIND THAT WE HAVE TO CONVERT ALL DENOMINATORS PART OF SUB PARTS OF WHOLE TRIGONOMETRIC EXPRESSIONS INTO NUMERATORS BY CHOOSING RIGHT LINE SEGMENT OR THROUGH REVERSING THE DOUPLETS OF GT STRINGS. THEN WE NEED TO COMPLETE ALL * TASKS(OPERATIONS) THEN ONLY WE CAN DO + - TASKS[OPERATIONS] GEOMETRICALLY OTHERWISE WE WILL DO WRONG INTERPRETATIONS OF GEOMETRY

Algebraic Structure of Geometrifying Trigonometry https://camo.githubusercontent.com/4039160357cbd661fcd8d531817b83895bb4039b/68747470733a2f2f7170682e66732e71756f726163646e2e6e65742f6d61696e2d71696d672d3233633363613537653636383063363537633966626666323133313330656335 We have defined Several operations of Geometrifying Trigonometry which looks similar to that of Vector Algebra but the definitions of Equality , Definitions of Summations , Definitions of Toe Tips are all very much different and too much rigid in case of Geometrifying Trigonometry.

Although we use Vector Algebra to model several real life problems , We can use Geometrifying Trigonometry for several mathematical modeling systems to generate alternative identity conditions , or several types of visualization problems or to model scenario analysis where space planning is necessary

We have checked that many of such operations are not completely cross verified yet in CAD programming.We are working with checking of Associativity for all 7 types of * operations and also for Other operations.We are trying to search best algorithms or redefinitions regarding PIVOT , ALIGN,SCALE TO FIT of ∞ operation where as we have seen through simulations that ∞/ and ∞\ works best

Geometrifying Trigonometry

Geometrifying Trigonometry

Geometrifying Trigonometry:[Cos(Θ)]^6 Representations

SanjoyNaths Conjecture:Geometrifying Trigonometry: Tips of Line Segment (1+(Sin(Θ))^2)^0.5 and (1+(Cos(Θ))^2)^0.5 forms Cyclic Quadrilateral

Corollary:Geometrifying Trigonometry: Cos(Θ) x Sin(Θ)=Sin(Θ) x Cos (Θ) has 25% Chance to become equal

Corollary:Geometrifying Trigonometry: Cosec(Θ) x Cosec(Θ) - Cot(Θ) x Cot(Θ) = 1 represents PHPH - PBPB=P

Corollary:Geometrifying Trigonometry:Sec(Θ) x Sec(Θ) - Tan(Θ) x Tan(Θ) =1 That means BHBH-BPBP=B (The initial line segment considered as 1)

Corollary:Geometrifying Trigonometry:Tan(Θ) x Tan (Θ) is not Tan Squared Theta but it is BP * BP

Corollary:Geometrifying Trigonometry: Cos(Θ) x Cos (Θ) is not Cos Squared Θ it is HB*HB the line segment

Corollary :Geometrifying Trigonometry:Sin(Θ) x Sin (Θ) is not Sin Squared Θ , It is HP*HP

Theorem 10: Geometrifying Trigonometry :There are 6 Types of Equality in Geometrifying Trigonometry.≡ , = , ≈ , Æ , æ and ₧

HEP Arrangement is a Locked Set

Theorem 9:Geometrifying Trigonometry :Need definition of 1 and 0 for eiπ+1=0 Which is defined here (The initial line segment considered as 1)

Theorem 8:Geometrifying Trigonometry: Number of defined Line segments in HEP Arrangement is rank of the System

Theorem 6 :Geometrifying Trigonometry:There are Two Fundamental Types of Trigonometric Expressions Simple and Compound

Theorem 3++:Geometrifying Trigonometry:Every of Predefined line segments Locked on Locked Set HEP Arrangement has two Trigonometric Ratios involved

Theorem 7:Geometrifying Trigonometry: Every Compound Douplet String of Geometrifying Trigonometry Does not mean Reciprocal of Given Expression.

Theorem 3+:Geometrifying Trigonometry:The Division of Trigonometric Expression means a Star(*) Operation in String Duplets

Theorem 3:Geometrifying Trigonometry:The Multiplication of Trigonometric Expressions means a Star(*) Operation in Strings Duplets

Theorem 1 : Geometrifying Trigonometry :Contiguity of Intermediate Dot Operation is Worthless Other than First and Last Alphabet

Theorem 2: Geometrifying Trigonometry:Reciprocal String (Duplet) for Simple Ratios under Dot Operation is Reverse of that String(Duplet)

Strings in Geometrifying Trigonometry and interpretations(Part 2 HEA or HEP)

Circles Fourier Series and Geometrifying trigonometry GT String Formal Systems

Strings in Geometrifying Trigonometry and interpretations(Part 1 with Dots and Stars)

Operators , Their actions and Interpretations on GT String

Interpretation of Pythagoras Theorem in Geometrifying Trigonometry HBHB+HPHP=H (Line segment H Itself )

37 Line Segments and 10 Points Identified to Symbol set of Geometrifying Trigonometric Formal System

Automated Symbolic formulation of operations and formations due to operations on GT Strings in Geometrifying Trigonometry

Implementation of First order ,Second Order And Higher Order logic through Stringology on Geometrifying Trigonometry Some more stringology with Geometrifying of Trigonometry Combinatorial properties of Geometrifying trigonometry Geometrifying Trigonometry Fundamental rules explained Some First words on Geometrifying trigonometry Geometrifying Trigonometry

https://geometrifyingtrigonometry.quora.com/Geometrifying-Trigonometry-Constructing-Unconstructable-Numbers https://geometrifyingtrigonometry.quora.com/Geometrifying-Trigonometry-namespace-Tekla-CopyGADrawing https://geometrifyingtrigonometry.quora.com/Concerns-in-Geometrifying-Trigonometry-and-the-Building-Informations-Modeling-Issues https://geometrifyingtrigonometry.quora.com/Concerns-Geometrifying-Trigonometry-Overlapped-Line-segments-Jungles https://geometrifyingtrigonometry.quora.com/Geometrifying-Trigonometry-public-class-TeklaDrawingAPIServices https://geometrifyingtrigonometry.quora.com/Geometrifying-Trigonometry https://geometrifyingtrigonometry.quora.com/Geometrifying-Trigonometry-Cos-%CE%98-6-Representations https://geometrifyingtrigonometry.quora.com/SanjoyNaths-Conjecture-Geometrifying-Trigonometry-Tips-of-Line-Segment-1+-Sin-%CE%98-2-0-5-and-1+-Cos-%CE%98-2-0-5-for https://geometrifyingtrigonometry.quora.com/Corollary-Geometrifying-Trigonometry-Cos-%CE%98-x-Sin-%CE%98-Sin-%CE%98-x-Cos-%CE%98-has-25-Chance-to-become-equal https://geometrifyingtrigonometry.quora.com/Corollary-Geometrifying-Trigonometry-Cosec-%CE%98-x-Cosec-%CE%98-Cot-%CE%98-x-Cot-%CE%98-1-represents-PH*PH-PB*PB-P https://geometrifyingtrigonometry.quora.com/Corollary-Geometrifying-Trigonometry-Sec-%CE%98-x-Sec-%CE%98-Tan-%CE%98-x-Tan-%CE%98-1-That-means-BH*BH-BP*BP-B-The-initial-line https://geometrifyingtrigonometry.quora.com/Corollary-Geometrifying-Trigonometry-Tan-%CE%98-x-Tan-%CE%98-is-not-Tan-Squared-Theta-but-it-is-BP-*-BP https://geometrifyingtrigonometry.quora.com/Corollary-Geometrifying-Trigonometry-Cos-%CE%98-x-Cos-%CE%98-is-not-Cos-Squared-%CE%98-it-is-HB*HB-the-line-segment https://geometrifyingtrigonometry.quora.com/Corollary-Geometrifying-Trigonometry-Sin-%CE%98-x-Sin-%CE%98-is-not-Sin-Squared-%CE%98-It-is-HP*HP https://geometrifyingtrigonometry.quora.com/Theorem-10-Geometrifying-Trigonometry-There-are-6-Types-of-Equality-in-Geometrifying-Trigonometry-%E2%89%A1-%E2%89%88-%C3%86-%C3%A6 https://geometrifyingtrigonometry.quora.com/HEP-Arrangement-is-a-Locked-Set https://geometrifyingtrigonometry.quora.com/Theorem-9-Geometrifying-Trigonometry-Need-definition-of-1-and-0-for-ei%CF%80+1-0-Which-is-defined-here-The-initial-line https://geometrifyingtrigonometry.quora.com/Theorem-8-Geometrifying-Trigonometry-Number-of-defined-Line-segments-in-HEP-Arrangement-is-rank-of-the-System https://geometrifyingtrigonometry.quora.com/Theorem-6-Geometrifying-Trigonometry-There-are-Two-Fundamental-Types-of-Trigonometric-Expressions-Simple-and-Compoun https://geometrifyingtrigonometry.quora.com/Theorem-3++-Geometrifying-Trigonometry-Every-of-Predefined-line-segments-Locked-on-Locked-Set-HEP-Arrangement-has-two https://geometrifyingtrigonometry.quora.com/Theorem-7-Geometrifying-Trigonometry-Every-Compound-Douplet-String-of-Geometrifying-Trigonometry-Does-not-mean-Recip https://geometrifyingtrigonometry.quora.com/Theorem-3+-Geometrifying-Trigonometry-The-Division-of-Trigonometric-Expression-means-a-Star-*-Operation-in-String-Du https://geometrifyingtrigonometry.quora.com/Theorem-3-Geometrifying-Trigonometry-The-Multiplication-of-Trigonometric-Expressions-means-a-Star-*-Operation-in-Str https://geometrifyingtrigonometry.quora.com/Theorem-1-Geometrifying-Trigonometry-Contiguity-of-Intermediate-Dot-Operation-is-Worthless-Other-than-First-and-La https://geometrifyingtrigonometry.quora.com/Theorem-2-Geometrifying-Trigonometry-Reciprocal-String-Duplet-for-Simple-Ratios-under-Dot-Operation-is-Reverse-of https://geometrifyingtrigonometry.quora.com/Strings-in-Geometrifying-Trigonometry-and-interpretations-Part-2-HEA-or-HEP https://geometrifyingtrigonometry.quora.com/Circles-Fourier-Series-and-Geometrifying-trigonometry-GT-String-Formal-Systems https://geometrifyingtrigonometry.quora.com/Strings-in-Geometrifying-Trigonometry-and-interpretations-Part-1-with-Dots-and-Stars https://geometrifyingtrigonometry.quora.com/Operators-Their-actions-and-Interpretations-on-GT-String https://geometrifyingtrigonometry.quora.com/Interpretation-of-Pythagoras-Theorem-in-Geometrifying-Trigonometry-HB*HB+HP*HP-H-Line-segment-H-Itself https://geometrifyingtrigonometry.quora.com/37-Line-Segments-and-10-Points-Identified-to-Symbol-set-of-Geometrifying-Trigonometric-Formal-System https://geometrifyingtrigonometry.quora.com/Automated-Symbolic-formulation-of-operations-and-formations-due-to-operations-on-GT-Strings-in-Geometrifying-Trigonom https://geometrifyingtrigonometry.quora.com/Implementation-of-First-order-Second-Order-And-Higher-Order-logic-through-Stringology-on-Geometrifying-Trigonometry https://geometrifyingtrigonometry.quora.com/Some-more-stringology-with-Geometrifying-of-Trigonometry https://geometrifyingtrigonometry.quora.com/Combinatorial-properties-of-Geometrifying-trigonometry https://geometrifyingtrigonometry.quora.com/Geometrifying-Trigonometry-Fundamental-rules-explained https://geometrifyingtrigonometry.quora.com/Some-First-words-on-Geometrifying-trigonometry https://geometrifyingtrigonometry.quora.com/Geometrifying-Trigonometry