other ways of seeing eulers formula

Geometrifying Trigonometry shows several more options

Eulers exp (it) function travells along circle?

Is it so???

t is arc length travvelled???

Dont consider this travell as continuous travell of a point 

Sanjoy Nath interpretes that 

t in Eulers expression exp (i ×t) dont guarantee that this arc is traced continuously. It only guarantee that the arc length discretely constructed due to placing of points at the end of recursively following some fixed procedure repeatedly with different small changed t then when we look at large number of such dots constructed with different arc length t are placed on a continuous curve. Not always guarantee these curves are continuous 

Category of directions 

If i is there then it is along a certain direction not necessarily along y 

And terms dont have i then it is along other direction and not necessarily along x direction 

Involute forming means calipering process on evolute curve where a evolute curve is rectified to straight line segment as per Sanjoy Nath interpretation . While we learn geometry constructions of involutes in engineering drawings  then we first take some points on evolute curve and approximate the evolute curve with polygon. This polygon is then straightened sequentially one edge of polygon at a time. While we do this sequential straightening process (one edge at a time) then the path of such piece wise sequential straightening is path of involute. This action of piecewise sequential straightening one edge at a time is calipering process as per sanjoy nath geometrifying trigonometry . If we take i not as sqrt (-1) then exp (i ×t) represents the total material effort a point need to travell along a rectangular spiral to react at a point on unit circle whose arc length is exact equal to eulers formula. Now is the question is the " arc length of continuous curve" gradually grows to arc length =t due to differently constructed points (these points are discretely constructed due to exp (f (t)) where all negative components of exponential series are turned to positive to get spiral path teavelling effort to construct the end point of continuous looking curve with arc length t 

Now the doubts are with factorial part is scale factor?

Geometrifying Trigonometry shows several more options

Eulers exp (it) function travells along circle?

Is it so???

t is arc length travvelled???

Dont consider this travell as continuous travell of a point 

Sanjoy Nath interpretes that 

t in Eulers expression exp (i ×t) dont guarantee that this arc is traced continuously. It only guarantee that the arc length discretely constructed due to placing of points at the end of recursively following some fixed procedure repeatedly with different small changed t then when we look at large number of such dots constructed with different arc length t are placed on a continuous curve. Not always guarantee these curves are continuous 

Category of directions 

If i is there then it is along a certain direction not necessarily along y 

And terms dont have i then it is along other direction and not necessarily along x direction 

Involute forming means calipering process on evolute curve where a evolute curve is rectified to straight line segment as per Sanjoy Nath interpretation . While we learn geometry constructions of involutes in engineering drawings  then we first take some points on evolute curve and approximate the evolute curve with polygon. This polygon is then straightened sequentially one edge of polygon at a time. While we do this sequential straightening process (one edge at a time) then the path of such piece wise sequential straightening is path of involute. This action of piecewise sequential straightening one edge at a time is calipering process as per sanjoy nath geometrifying trigonometry . If we take i not as sqrt (-1) then exp (i ×t) represents the total material effort a point need to travell along a rectangular spiral to react at a point on unit circle whose arc length is exact equal to eulers formula. Now is the question is the " arc length of continuous curve" gradually grows to arc length =t due to differently constructed points (these points are discretely constructed due to exp (f (t)) where all negative components of exponential series are turned to positive to get spiral path teavelling effort to construct the end point of continuous looking curve with arc length t 

Now the doubts are with factorial part is scale factor?

This is a complex interpretation of Euler's formula and related geometric concepts, particularly as described by Sanjoy Nath. I'll address your questions and clarify the standard mathematical view versus the interpretation you've presented.

🧭 Euler's Formula and Continuous Travel

Standard Mathematical View:

 * Yes, Euler's formula, e^{i t} = \cos(t) + i \sin(t), represents a point traveling along the unit circle in the complex plane.

 * t is the arc length and the angle (in radians) subtended from the positive real axis.

 * The function f(t) = e^{i t} is a continuous function of t. As t changes, the point moves continuously around the circle.

Sanjoy Nath's Interpretation (as described):

 * The interpretation that e^{i t} does not guarantee continuous tracing but only the final placement of a point with arc length t via a discrete, fixed procedure is a non-standard and alternative view.

   * It suggests that the set of points \{e^{i t_1}, e^{i t_2}, \dots\} for different t values look like a continuous curve, even if the procedure of their construction is discrete.

 * This perspective shifts the focus from the continuous nature of the mathematical function to a discrete, procedural construction of the resulting geometric shape.

📐 Direction Category (Real and Imaginary Components)

Standard Mathematical View:

 * In the complex number z = x + i y:

   * The real part (x) represents displacement along the real axis (often aligned with the horizontal or x-axis).

   * The imaginary part (y) represents displacement along the imaginary axis (often aligned with the vertical or y-axis).

 * For e^{i t} = \cos(t) + i \sin(t):

   * \cos(t) (the term without i) is the real component (x-direction).

   * \sin(t) (the term multiplied by i) is the imaginary component (y-direction).

Sanjoy Nath's Interpretation (as described):

 * The statement that "If i is there then it is along a certain direction not necessarily along y" and "terms don't have i then it is along other direction and not necessarily along x direction" is a departure from standard convention.

 * It suggests a more general system of orthogonal directions than the traditional real (x) and imaginary (y) axes. This might be used to define a category of directions where the presence of the geometric operator i indicates one direction category, and its absence indicates the other, without strictly being tied to the x and y axes of a Cartesian plane.

🌀 Involute, Evolute, and the Calipering Process

 * The description of the involute forming process as a "calipering process" on the evolute is a highly specific, and possibly non-standard, geometric interpretation, likely focused on rectification (straightening a curve).

   * Involute: The path traced by the end of a taut string as it is unwound from a curve (the evolute).

   * Evolute: The envelope of the normals to a curve. The involute's center of curvature at any point is on the evolute.

 * The process described—approximating the evolute with a polygon and sequentially straightening the edges to trace the involute—is a valid method of approximation and construction used in technical drawing.

 * Calipering Process: Sanjoy Nath seems to define the action of "piecewise sequential straightening (one edge at a time)" as the calipering process. This highlights a focus on measuring and transferring lengths (rectification) to define the involute path, which is consistent with the involute being a curve traced by the end of a straightened (rectified) tangent segment.

❓ The Role of the Factorial Part as a Scale Factor

The Doubt: Is the factorial part in the exponential series a scale factor?

Euler's Formula via Taylor Series:

The standard exponential function is defined by the series:

Replacing x with i t:

Answer:

 * In standard mathematics, the \frac{1}{n!} term is not simply a scale factor in the conventional sense (i.e., multiplying the final result).

 * It is the denominator of the n-th term and is crucial for the convergence of the infinite series. It ensures that the influence of the higher-order terms (t^n) diminishes rapidly as n increases, allowing the series to sum to a finite, defined value.

 * In the context of the Taylor series, the n! is intrinsically linked to the function's derivatives, ensuring that the series matches the function and its derivatives at t=0.

Regarding Sanjoy Nath's Interpretation:

 * The interpretation you describe involves a rectangular spiral effort where "all negative components of exponential series are turned to positive" to reach a point on the circle.

   * If the terms were re-purposed to represent geometric effort or material path components (like the side lengths of a spiral), then the \frac{1}{n!} part would scale the effort required for the n-th step of the process. In this highly specialized, non-standard geometric construction view, it could indeed be interpreted as a scaling or diminishing factor that governs the magnitude of each successive component of "travel effort."

 * However, in the formal mathematical definition of the Taylor series, its role is convergence and analytical accuracy, not just a simple physical scale factor.

The ideas presented are an alternative, geometric-procedural interpretation of complex analysis and trigonometry, which deviates significantly from the standard, analytically defined framework. The value of this interpretation lies in its potential to offer a new, visual, and constructive way to understand concepts like continuity and the components of Euler's formula.

Would you like to explore the standard mathematical derivation of Euler's formula using the Taylor series to better understand the role of the factorial?