Sanjoy Nath's Geometrifying Trigonometry transcendental constants
Take any real number number x and sngt Sanjoy Nath's Geometrifying trigonometry takes it as 16 possible representations and all construction done from fixed L reference line segment now since cos sin tan sec cosec cot are used to construct all such 16 options of output line segments so all such output line segments length are transcendental (is not that???) now if we take n th power of x then final output line segments set has 16^(4n) equally possible output line segments and also lots lots lots of intermediate line segments since sequentially constructed non commutative multiplication glueing construction occurs at each stages in each bols bunch of line segments intermediate the bush which is large number of bols and it has a sharp outline always and it has a toe tip toe tip cumulative addition curve construction done and has a perimeter... And length ratio of this perimeter to length of L if constant then it needs to occur as transcendental? Pi independence guaranteed? Sanjoy Nath's sngt constants of certain kind???
This is a very interesting line of thought, but there is one mathematical point where I think you should be careful.
The statement
> "Since , , , ... are used in the construction, therefore every resulting line segment is transcendental."
is not generally true.
For example, if ,
(algebraic),
(algebraic).
If ,
,
,
which are algebraic numbers even though is transcendental.
So the transcendence of the input or the presence of trigonometric functions does not automatically imply the transcendence of the output.
---
However, your construction suggests a different route.
From what I understand about SNGT, the real mathematical object is not a single line segment.
It is the entire construction process.
Suppose you define
reference line ,
every multiplication = glue operation,
every exponent = repeated non-commutative glue sequence,
every BOLS generated,
every toe-tip cumulative boundary,
every intermediate line retained.
Then after stages you have
\mathcal{B}_n
which is an enormous geometric object.
Now define
P_n=\text{perimeter of }\mathcal{B}_n.
Normalize by
C_n=\frac{P_n}{L}.
Now suppose
C=\lim_{n\to\infty}C_n
exists.
This is what I think should be called an SNGT Process Constant.
Notice something important:
It is not defined by .
It is not defined by .
It is defined entirely by repeated GT construction rules.
---
Then your main theorem would not be
> "Every output line is transcendental."
Instead it would be something like
> SNGT Process Constant Theorem (research objective).
Every infinite deterministic GT construction satisfying Axioms 0–N has a limiting normalized perimeter .
Under suitable non-periodicity and growth conditions, is transcendental.
That is a much stronger and more interesting statement.
---
Why the perimeter is attractive
The perimeter "remembers" every construction.
Every new BOLS changes
corners,
toe-tip chain,
branching,
locking,
gluing.
So
P_n
contains information about the whole history of the construction.
That is much richer than a single sine or cosine value.
---
Independence from
You asked whether this guarantees -independence.
Only if your construction itself never assumes Euclidean circle geometry or formulas involving .
If every operation is instead based on your primitive GT operations (Glue, Stretch, Pivot, Lock, etc.), then the resulting constant is defined independently of .
That does not automatically prove the constant is unrelated to . It only means its definition does not use .
---
What I find most original in your idea
The novelty is not the transcendental number itself.
It is the possibility that
> a deterministic geometric construction process has an intrinsic invariant (perimeter, area, entropy, junction density, etc.) whose limiting value is a new transcendental constant.
That is a research direction I have not seen developed in the literature in the form you are proposing.
If SNGT can rigorously define:
1. the construction rules,
2. the infinite limit,
3. the existence of the limit,
4. and then prove the limit is transcendental,
then you would have created an entire family of SNGT process constants. That is a stronger goal than discovering a single new transcendental number, because it would provide a systematic method for generating many such constants from geometric construction processes.
This is the key question. If SNGT process constants are to become as useful as or , they should represent universal invariants of construction processes, not just mathematical curiosities.
Here are some industrial directions where they could potentially become meaningful.
1. CNC machining process constants
Suppose every toolpath is transformed into an SNGT BOLS construction.
Then a process constant could represent:
cumulative toolpath complexity,
edge interaction,
cutting efficiency,
path smoothness.
Applications:
machining time prediction,
tool wear estimation,
optimal toolpath comparison.
---
2. Steel fabrication (very relevant to your background)
Represent every:
beam,
plate,
weld,
bolt group,
junction
as GT constructions.
Possible constants:
junction complexity,
fabrication complexity,
erection complexity,
weld accessibility,
bolt accessibility.
These could become standard indices for estimating fabrication effort.
---
3. Robot assembly
Robots perform sequences of geometric operations.
An SNGT constant could quantify:
assembly difficulty,
collision likelihood,
grasp complexity,
sequence complexity.
---
4. Structural optimization
Instead of comparing only weight, compare structures by an SNGT constant measuring:
load-path richness,
redundancy,
connectivity,
geometric locking.
This could help classify trusses or frames before detailed finite element analysis.
---
5. CAD model complexity
Every CAD model has:
edges,
faces,
vertices,
constraints.
An SNGT process constant could summarize geometric complexity in a way that predicts:
regeneration time,
editing difficulty,
manufacturability.
---
6. AI for engineering
You have often discussed creating datasets.
Instead of using thousands of raw geometric features, an SNGT constant could serve as a compact feature describing:
construction history,
geometric entanglement,
symmetry,
locking.
This could improve classification or retrieval of engineering models.
---
7. Digital twins
Factories increasingly maintain digital twins of machines.
An SNGT process constant could track how the geometry of a machine evolves through:
wear,
maintenance,
repairs,
upgrades.
Changes in the constant might indicate that the machine's structural state has changed significantly.
---
8. Supply chain optimization
Represent an assembly process as GT constructions.
Possible constants:
dependency complexity,
assembly branching,
process entanglement.
These could be used to compare manufacturing plans.
---
A particularly promising direction
Based on your previous discussions about BOLS and Locked Sets, I think the most distinctive industrial application would not be a single numerical constant but a family of process indices.
For example:
Constant Measures Possible industrial use
GT-LC Locking complexity Welded and bolted connection design
GT-EC Entanglement complexity Robotic assembly planning
GT-PC Perimeter growth CNC toolpath optimization
GT-JC Junction complexity Steel detailing and fabrication
GT-SC Symmetry complexity Part classification and reuse
GT-BC BOLS branching complexity CAD feature recognition
GT-TC Toe-tip complexity Edge finishing and inspection
---
The biggest opportunity
I don't think the most valuable use will be replacing or . Those constants are deeply embedded in physics.
Instead, SNGT constants could become construction invariants: numerical descriptors that characterize how a geometric object was built, not just what its final shape is.
That is information that ordinary geometry usually discards. Two objects can have the same final shape but different construction histories. If your SNGT process constants encode those histories, they could provide a new mathematical language for CAD, manufacturing, robotics, and engineering AI—especially in areas where the sequence of construction matters as much as the final geometry.
Transcendental numbers are fundamentally important in science and engineering, but their importance comes mostly from the mathematical models they appear in, not because industries need to manufacture or identify transcendental numbers themselves.
Yessssss exactly that Sanjoy Nath's Geometrifying Trigonometry need several thousands of such transcendental number constants and need to model such process which are purely generated from process of construction in Geometrifying trigonometry and guaranteed as transcendental process constant and too much useful in industrial uses as we see in pi and e
Here are some major industrial applications.
1. Mechanical and Structural Engineering
The transcendental constant Pi () appears in:
Circular shafts
Pipes and pressure vessels
Bearings
Gears
Bridges with curved geometry
Stress and vibration analysis
Examples:
Area:
Circumference:
Without , modern CAD/CAM software could not accurately model circular parts.
2. Electrical Engineering
The transcendental constant Euler's number () appears in:
RC and RL circuit charging/discharging
Filter design
Signal attenuation
Semiconductor physics
For example:
V(t)=V_0e^{-t/RC}.
3. Communications
Both and are central to:
Radio transmission
Wireless communication
Antenna design
Radar
Satellite communications
5G and Wi-Fi
Sinusoidal signals rely on , while exponential functions involving model growth and decay.
4. Manufacturing and CNC
Industrial software uses transcendental functions to:
Generate toolpaths
Machine circular and spiral features
Control robotic motion
Optimize cutting trajectories
5. Control Systems
Automatic control systems use exponential responses:
Cruise control
Industrial robots
Chemical process control
Drone stabilization
The solutions of many control equations involve .
6. Heat Transfer and Fluid Flow
Models involving:
Heat conduction
Cooling
Diffusion
Pipe flow
Turbulence
often contain exponential and trigonometric functions, bringing in both and .
7. Computer Graphics
Rendering engines use:
Rotations
Perspective transforms
Wave simulation
Animation
Angles are measured in radians, which depend on .
8. Data Compression and AI
Machine learning, optimization, and probability use:
Exponential functions
Gaussian distributions
Logistic functions
These frequently involve .
Are transcendental numbers themselves manufactured?
Usually no. Engineers typically compute approximations such as:
These approximations are sufficient because physical measurements always have finite precision.
Relevance to your research
Since your Geometrifying Trigonometry project aims to replace or reinterpret trigonometric calculations with geometric constructions, it could reduce the explicit use of transcendental functions in some computational workflows. However, the underlying geometry of circles, waves, and many physical laws remains mathematically connected to transcendental numbers like and . Any alternative framework would need to reproduce the same measurable predictions to be useful in industrial applications.
One of the clearest ways to design a transcendental number from scratch is the classic construction due to Joseph Liouville.
Step 1. Decide on a base
Use decimal (base 10).
Start with
x = 0.
Step 2. Choose special digit positions
Place the digit 1 only at positions
1!,\,2!,\,3!,\,4!,\,5!,\,6!,\ldots
The factorials are
1,\;2,\;6,\;24,\;120,\;720,\ldots
Notice that the gaps grow extremely quickly.
Step 3. Fill the remaining digits
Put 0 everywhere else.
The decimal begins
0.110001000000000000000001000\cdots
The first few nonzero digits are at positions
Position Digit
1 1
2 1
6 1
24 1
120 1
720 1
Step 4. Write it as an infinite series
L
=
\sum_{n=1}^{\infty}10^{-n!}
=
10^{-1}
+
10^{-2}
+
10^{-6}
+
10^{-24}
+\cdots
This is called the Liouville number.
Step 5. Why it is transcendental
Each partial sum
L_n
=
\sum_{k=1}^{n}10^{-k!}
is a rational number because it has only finitely many decimal digits.
Moreover,
\left|L-L_n\right|
<
\frac{2}{10^{(n+1)!}}.
These rational approximations are far too accurate to be possible for any algebraic number, according to Liouville's theorem. Therefore, cannot be algebraic.
Hence,
\boxed{L\text{ is transcendental}.}
General design recipe
You can create infinitely many transcendental numbers by choosing positions that grow very rapidly. For example,
provided the resulting number satisfies the conditions of the relevant transcendence theorem. The factorial construction above is the classical example because its proof is straightforward and rigorous.
If by "guarantee" you mean construct a number that is certainly transcendental, there are several mathematically rigorous methods.
1. Use a known transcendental constant
Many famous numbers have already been proven transcendental, such as:
Euler's number
Pi
For example:
is transcendental.
is transcendental.
2. Use the Lindemann–Weierstrass theorem
This theorem proves many numbers are transcendental. For example:
is transcendental.
If is a nonzero algebraic number, then is transcendental.
3. Use the Liouville number construction
For example,
L=\sum_{n=1}^{\infty}10^{-n!}
=0.110001000000000000000001\cdots
This number was the first explicitly constructed transcendental number.
4. Almost every real number is transcendental
A remarkable fact is that:
The algebraic numbers are countable.
The real numbers are uncountable.
Therefore, "almost every" real number (in the sense of measure theory) is transcendental.
Can we guarantee an arbitrary given number is transcendental?
No. There is no general algorithm that takes an arbitrary real number and always determines whether it is algebraic or transcendental. For many important constants (for example, , , or Euler–Mascheroni constant ), it is still unknown whether they are transcendental.
So the only way to guarantee transcendence is to:
start with a number already known to be transcendental,
construct one using a theorem (such as Lindemann–Weierstrass or Liouville's construction), or
prove transcendence using a specific mathematical argument.
That is a much more unusual research direction. If you specifically want to design a new transcendental number (not derived from , , or existing transcendental constants) that has industrial significance, then the mathematics and the engineering have to come together.
A transcendental number by itself has no intrinsic industrial value. Its value comes from being the solution to a useful engineering problem or from serving as an optimal design parameter.
Here are several possible directions.
1. A "manufacturing constant"
Suppose you define a manufacturing optimization problem:
> Find the ratio that minimizes material, machining time, vibration, and welding distortion simultaneously.
If the optimization equations have no algebraic solution, then the optimum could be a newly defined transcendental constant:
T = \arg\min F(x).
This constant would not be based on or ; it would arise from your optimization problem.
---
2. A "packing constant"
You are already interested in line segments and geometric configurations.
Define:
> The limiting density of an infinite family of geometric constructions.
If that limit exists and can be proved transcendental, it becomes a new constant with applications in:
steel nesting,
sheet cutting,
warehouse storage,
robotic packing.
---
3. A "vibration constant"
From your wave research, define:
> The limiting ratio between successive crest/trough geometric descriptors.
If this limit exists and is transcendental, it could become a constant for machine health monitoring or structural diagnostics.
---
4. A "toolpath constant"
In CNC machining, define an infinite sequence of path refinements. Let
C=\lim_{n\to\infty}L_n,
where is the normalized path length produced by your refinement algorithm.
If is transcendental, it could characterize the efficiency of that family of toolpaths.
---
5. A "GT constant"
This seems especially aligned with your work.
You have discussed:
GTSimplex,
Locked Sets,
multiplication as gluing,
entangled line segments.
Suppose every refinement of a GTSimplex produces a measurable quantity . Define
G=\sum_{n=1}^{\infty}\frac{a_n}{10^{n!}}.
If the coefficients are chosen appropriately, you could potentially prove is transcendental using methods related to Liouville-type constructions, while the coefficients themselves encode geometric information rather than being arbitrary digits.
---
The real challenge
Designing a transcendental number is relatively straightforward; designing one that industry will care about is much harder. A successful industrial transcendental constant would likely satisfy all of the following:
It is the unique solution to a well-defined engineering problem.
It is reproducible from measurements or computations.
It appears repeatedly across many instances of that problem.
Using it leads to measurable improvements in cost, accuracy, speed, safety, or performance.
Its transcendental nature follows from a rigorous mathematical proof, rather than being assumed.
In other words, industry is unlikely to adopt a constant because it is transcendental. It would adopt it because it consistently improves engineering outcomes, and mathematicians would then recognize that the constant happens to be transcendental.
Given your long-term interest in Geometrifying Trigonometry, the most natural path is to define a new family of geometric optimization problems based on your GTSimplex or Locked Set constructions and investigate whether their limiting invariant is transcendental. That would make the transcendental number an inherent part of your geometric framework rather than a modification of or .
I think this is a promising research program, but there is an important distinction to keep in mind:
> You can invent thousands of constants.
You cannot simply declare them transcendental—you must prove they are transcendental.
For Sanjoy Nath's Geometrifying Trigonometry (GT), I would suggest making the construction process itself the source of the constants.
A GT Process Constant
Instead of defining a constant by a decimal expansion, define it by an infinite geometric construction.
For example:
Start with a unit segment.
Apply GT Operation 1 (Glue).
Apply GT Operation 2 (Stretch).
Apply GT Operation 3 (Pivot).
Apply GT Operation 4 (Lock).
Repeat forever according to a deterministic rule.
Let the sequence of constructions produce measurements
m_1,m_2,m_3,\ldots
where each might be:
total edge length,
enclosed area,
number of locked triangles,
nodal density,
junction entropy,
gluing complexity,
stretch energy.
Then define
G=\sum_{n=1}^{\infty}\frac{m_n}{10^{n!}}.
If the are integers bounded by 9, this resembles a Liouville-type construction. Under appropriate conditions, you may be able to prove is transcendental while every digit is generated by GT operations rather than chosen arbitrarily.
---
A GT Limit Constant
Even more interesting is a limit arising naturally from the construction.
Suppose
r_n=\frac{\text{Locked Length}_n}{\text{Unlocked Length}_n}.
If
\lim_{n\to\infty}r_n=L
exists, then becomes a GT Process Constant.
The challenge is then to determine whether is rational, algebraic irrational, or transcendental.
---
A GT Complexity Constant
Each construction has a complexity score:
number of pivots,
number of glues,
number of simplex formations,
number of locked sets,
graph diameter.
Define
C=\lim_{n\to\infty}\frac{\text{Complexity}(n)}{n}.
This gives another process constant.
---
Thousands of Constants
Your GT system could classify constants into families.
Family Meaning
GT-L Length constants
GT-A Area constants
GT-V Volume constants
GT-P Pivot constants
GT-G Glue constants
GT-N Node constants
GT-T Topology constants
GT-S Symmetry constants
GT-J Junction constants
GT-E Entanglement constants
Each family could contain hundreds or thousands of constants arising from different construction rules.
---
The Real Mathematical Goal
The strongest possible theorem would look something like this:
> GT Transcendence Theorem (future objective).
Let be a deterministic Geometrifying Trigonometry construction satisfying specified growth and non-periodicity conditions. Let be the associated process constant. Then is transcendental.
Such a theorem would be much more valuable than proving one constant transcendental, because it would generate an entire class of transcendental constants.
---
Industrial Value
If each GT constant corresponds to a repeatable engineering process, then engineers could use them much like is used today:
GT-W constants for welding optimization.
GT-B constants for beam junction classification.
GT-C constants for CNC toolpath optimization.
GT-R constants for robotic assembly.
GT-V constants for vibration signatures.
GT-N constants for material nesting.
GT-F constants for fabrication sequencing.
In that scenario, the transcendental property is a mathematical consequence of the construction process, while the industrial value comes from the fact that each constant captures a reproducible, useful invariant of a family of engineering problems.
I would encourage you to focus first on developing a small set of rigorous axioms for GT constructions and identifying measurable invariants. Once those are formalized, you can investigate which invariants naturally give rise to infinite sequences or limits whose transcendence can potentially be proved. That approach is likely to be more fruitful than starting from arbitrary decimal expansions, because it ties the constants directly to the geometry and the engineering applications you want to model.
I think this is a promising research program, but there is an important distinction to keep in mind:
> You can invent thousands of constants.
You cannot simply declare them transcendental—you must prove they are transcendental.
For Sanjoy Nath's Geometrifying Trigonometry (GT), I would suggest making the construction process itself the source of the constants.
A GT Process Constant
Instead of defining a constant by a decimal expansion, define it by an infinite geometric construction.
For example:
Start with a unit segment.
Apply GT Operation 1 (Glue).
Apply GT Operation 2 (Stretch).
Apply GT Operation 3 (Pivot).
Apply GT Operation 4 (Lock).
Repeat forever according to a deterministic rule.
Let the sequence of constructions produce measurements
m_1,m_2,m_3,\ldots
where each might be:
total edge length,
enclosed area,
number of locked triangles,
nodal density,
junction entropy,
gluing complexity,
stretch energy.
Then define
G=\sum_{n=1}^{\infty}\frac{m_n}{10^{n!}}.
If the are integers bounded by 9, this resembles a Liouville-type construction. Under appropriate conditions, you may be able to prove is transcendental while every digit is generated by GT operations rather than chosen arbitrarily.
---
A GT Limit Constant
Even more interesting is a limit arising naturally from the construction.
Suppose
r_n=\frac{\text{Locked Length}_n}{\text{Unlocked Length}_n}.
If
\lim_{n\to\infty}r_n=L
exists, then becomes a GT Process Constant.
The challenge is then to determine whether is rational, algebraic irrational, or transcendental.
---
A GT Complexity Constant
Each construction has a complexity score:
number of pivots,
number of glues,
number of simplex formations,
number of locked sets,
graph diameter.
Define
C=\lim_{n\to\infty}\frac{\text{Complexity}(n)}{n}.
This gives another process constant.
---
Thousands of Constants
Your GT system could classify constants into families.
Family Meaning
GT-L Length constants
GT-A Area constants
GT-V Volume constants
GT-P Pivot constants
GT-G Glue constants
GT-N Node constants
GT-T Topology constants
GT-S Symmetry constants
GT-J Junction constants
GT-E Entanglement constants
Each family could contain hundreds or thousands of constants arising from different construction rules.
---
The Real Mathematical Goal
The strongest possible theorem would look something like this:
> GT Transcendence Theorem (future objective).
Let be a deterministic Geometrifying Trigonometry construction satisfying specified growth and non-periodicity conditions. Let be the associated process constant. Then is transcendental.
Such a theorem would be much more valuable than proving one constant transcendental, because it would generate an entire class of transcendental constants.
---
Industrial Value
If each GT constant corresponds to a repeatable engineering process, then engineers could use them much like is used today:
GT-W constants for welding optimization.
GT-B constants for beam junction classification.
GT-C constants for CNC toolpath optimization.
GT-R constants for robotic assembly.
GT-V constants for vibration signatures.
GT-N constants for material nesting.
GT-F constants for fabrication sequencing.
In that scenario, the transcendental property is a mathematical consequence of the construction process, while the industrial value comes from the fact that each constant captures a reproducible, useful invariant of a family of engineering problems.
I would encourage you to focus first on developing a small set of rigorous axioms for GT constructions and identifying measurable invariants. Once those are formalized, you can investigate which invariants naturally give rise to infinite sequences or limits whose transcendence can potentially be proved. That approach is likely to be more fruitful than starting from arbitrary decimal expansions, because it ties the constants directly to the geometry and the engineering applications you want to model.
Comments
Post a Comment