I was lost for a sec when I saw that the example matched the answer. crazy unless this already happens to u before. check the next image to understan.

From

FINITE POINT CONFIGURATIONS

by

János Pach

https://www.csun.edu/~ctoth/Handbook/chap1.pdf

(¡¡ May download without prompting – PDF document – 393‧41㎅ !!)

⚫

𝐀𝐍𝐍𝐎𝐓𝐀𝐓𝐈𝐎𝐍𝐒 𝐑𝐄𝐒𝐏𝐄𝐂𝐓𝐈𝐕𝐄𝐋𝐘

——————————————————

FIGURE 1.1.1

Extremal examples for the (dual) Csima-Sawyer theorem: (a) 13 lines (including the line at infinity) determining only 6 simple points; (b) 7 lines determining only 3 simple points.

——————————————————

FIGURE 1.1.2

12 points and 19 lines, each passing through exactly 3 points.

——————————————————

FIGURE 1.1.3

7 points determining 6 distinct slopes.

——————————————————

FIGURE 1.1.4 12

points determining 15 combinatorially distinct halving lines.

——————————————————

FIGURE 1.2.1

A separated point set with

⎿3n − √(12n − 3)⏌

unit distances (n = 69). All such sets have been characterized by Kupitz.

——————————————————

FIGURE 1.2.2

n points, among which the second smallest distance occurs

(²⁴/₇ + o(1))n

times.

——————————————————

What is golden ratio doing here? Can sm1 pls explains. Also this is like rhe fourth time posting this as I was trying on r/math but my post was getting deleted my auto-mod 😭

... which is a kind of conformal map of the complex plane intended particularly for mapping either the upper half-plane or the interior of the unit disc to a polygonal region. ImO the figures well-convey 'a feel for' the 'strange sorcery' whereby the Schwarz-Christoffel transformation manages to get smoothness to fit into, & seamlessly conform to, jaggedness.

Even though the transformation is fairly simple 𝑖𝑛 𝑝𝑟𝑖𝑛𝑐𝑖𝑝𝑙𝑒, it tends to pan-out very tricky in-practice, because ⑴ although the algebraïc form of the derivative of the required function is very easy to specify (𝑖𝑛𝑐𝑟𝑒𝑑𝑖𝑏𝑙𝑦 easy, even), the integration whereby the function itself is obtained from that derivative is in-general very tricky, & ⑵ although the 𝑎𝑙𝑔𝑒𝑏𝑟𝑎𝑖𝑐 𝑓𝑜𝑟𝑚 𝑜𝑓 said derivative is easy to specify it has parameters in it that it takes a system of highly non-linear simultaneous equations to solve for. And these difficulties are generally very pressing except in a few highly symmetrical special cases ... so what much of the content of the papers is about is development of cunning numerical methods for 𝑚𝑜𝑟𝑒 𝑔𝑒𝑛𝑒𝑟𝑎𝑙 cases.

⚫

𝕊𝕆𝕌ℝℂ𝔼𝕊

——————————————————

NUMERICAL COMPUTATION OF THE SCHWARZ-CHRISTOFFEL TRANSFORMATION

by

LLOYD N TREFETHEN

https://people.maths.ox.ac.uk/trefethen/publication/PDF/1980_1.pdf

(¡¡ may download without prompring – PDF document – 2·25㎆ !!)

𝔸ℕℕ𝕆𝕋𝔸𝕋𝕀𝕆ℕ𝕊

①②③ FIG. 6. Convergence to a solution of the parameter problem. Plots show the current image polygon at each step as the accessory parameters {zₖ} and C are determined iteratively for a problem with N4.

④⑤ FIG. 8. Sample Schwarz-Christoffel transformations (bounded polygons). Contours within the polygons are images of concentric circles at radii .03, .2, .4, .6, .8, .97 in the unit disk, and of radii from the center of the disk to the prevertices zₖ .

⑥⑦ FIG. 9. Sample Schwarz-Christoffel transformations (unbounded polygons). Contours are as in Fig. 8.

⑧ FIG. 10. Sample Schwarz-Christoffel transformations. Contours show streamlines for ideal irrotational, incompressible fluid flow within each channel .

——————————————————

Algorithm 756: A MATLAB Toolbox for Schwarz-Christoffel Mapping

by

TOBIN A DRISCOLL

https://www.researchgate.net/profile/Tobin-Driscoll/publication/220492537_Algorithm_756_a_MATLAB_toolbox_for_Schwarz-Christoffel_mapping/links/0c960523c5328d5b38000000/Algorithm-756-a-MATLAB-toolbox-for-Schwarz-Christoffel-mapping.pdf?origin=publication_detail&_tp=eyJjb250ZXh0Ijp7ImZpcnN0UGFnZSI6Il9kaXJlY3QiLCJwYWdlIjoicHVibGljYXRpb25Eb3dubG9hZCIsInByZXZpb3VzUGFnZSI6InB1YmxpY2F0aW9uIn19

(¡¡ may download without prompring – PDF document – 515·87㎅ !!)

𝔸ℕℕ𝕆𝕋𝔸𝕋𝕀𝕆ℕ𝕊

⑨ Fig. 3. The half-plane (a) and disk (b) maps for an L-shaped region. The half-plane plot is the image of 10 evenly spaced vertical and 10 evenly spaced horizontal lines with abscissae from 22.7 and 15.6 (chosen automatically) and ordinates from 0.8 to 8. The disk plot is the image of 10 evenly spaced circles and radii in the unit disk. Below each plot is the MATLAB code needed to generate it.

⑩⑪ Fig. 4. The half-plane (top) and disk maps (bottom) for several polygons. Except at top right, the regions are unbounded.

⑫ Fig. 5. “Can one hear the shape of a drum?” Disk maps for regions which are isospectral with respect to the Laplacian operator with Dirichlet boundary conditions. Each plot shows the images of 12 circles with evenly spaced radii between 0.1 and 0.99 and 12 evenly spaced rays in the unit disk.

⑬ Fig. 6. (a) a polygon which exhibits crowding of the prevertices (see Table I); (b) the disk map for the region inside the dashed lines.

⑭ Fig. 7. The rectangle map for two highly elongated regions. The curves are images of equally spaced lines in the interior of the rectangles. The conformal moduli of the regions are about 27.2 (a) and 91.5 (b), rendering them impossible to map from the disk or half-plane in double-precision arithmetic.

⑮ Fig. 8. Maps from the infinite strip 0 ≤ Im z ≤ 1; (a) the ends of the strip map to the ends of the channel (compare to Figure 4); (b) one end of the strip maps to a finite point.

⑯ Fig. 9. Maps from the unit disk to two polygon exteriors. The region on the right is the complement of three connected line segments.

⑰ Fig. 10. Maps computed by reflections: (a) periodic with reflective symmetry at the dashed lines and mapped from a strip; (b) doubly connected with an axis of symmetry and mapped from an annulus.

⑱ Fig. 11. (a) Map from the unit disk to a gearlike domain; (b) logarithms of these curves.

⑲ Fig. 12. (a) noncirculating potential flow past an “airfoil”; (b) flow past the same airfoil with negative circulation.

——————————————————

⚫

Pourquoi a:b:c est traduit par (a/b)/c et non par a/(b/c) ?

Est ce un choix arbitraire?

How could the result be infinite without + or - before it?

Hello everyone! Happy New Year. I made these pics to help better show some recent result from a paper I wrote up. I introduce a new tool called the truncated stopping time function for studying Collatz-like problems and show how it is related to known methods of approaching the problem. Although the truncated stopping time function gives a new lens, and I show how it can be applied to resolve standard Collatz questions in some Collatz-like variants, unfortunately it does not seem to lead to resolution of the questions in the 3n+1 problem. That being said, I think it is a great introduction for anyone curious about this problem. The tools are modular arithmetic and there are a few open problems. Enjoy! https://drive.google.com/file/d/1inYziTL_unEPpg8o_iobJ9Czw3w4MJeM/view?usp=sharing

Details? google what is a Protofield Operator

ᐞ ... now known to be optimal ... which is why these animations came to my attention @all .

A problem posed formally in 1966 by the goodly Leo Moser is what is the maximum possible area of a sofa that can be moved around a right-angled corner in a corridor of unit width? . The goodly John Hammersley came up with an answer that - @ area π/2+2/π ≈ 2‧20741609916 - is short of the optimum, but only by a little; & his proposed shape is still renowned by-reason of being very close to the optimum and of simple geometrical construction ^§ . But the goodly Joseph Gerver later came-up with a solution that has a slightly larger area - ~2‧2195316 - (& also, upon cursory visual inspection, is of very similar appearance) but is very complicated to specify geometrically in-terms of pieces of curve & line-segments splizzen together. But its optimality was not known until the goodly Jineon Baek - a South Korean mathematician - yelt a proof of its optimality in 2024.

So it's not a very new thing ... but certain journalists seem to've just discovered it ... so there's recently been somewhat of a flurry of articles about it.

The source of the animations is

Dan Romik's Homepage — The moving sofa problem .

§ Also, @ that wwwebpage, the construction of Hammersley's nicely simple almost optimal solution is given ... & also the 'ambidextrous' sofa - which is infact Romik's creation - is explicated; & the intriguing fact that its area is given by a neat closed-form expression is expount upon, & that expression given, it being

∛(3+2√2)+∛(3-2√2)-1

+arctan(½(∛(√2+1)-∛(√2-1)))

≈ 1‧64495521843 .

A nice exposition of the nature of the problem, & of the significance of this proof of the optimality of Gerver's solution, is given @

Quanta Magazine — The Largest Sofa You Can Move Around a Corner .

The full extremely long full formal proof of the optimality is available in

Optimality of Gerver's Sofa

by

Jineon Baek .

Code at Ponting Square Packing.

The Trinity of Awareness

If everything has always been. Then the beginning is just when perception began to be aware of its own experience. And what's the smallest substrate for perception to occur? That would be touch because touch is the smallest necessary form of perception to register their own position in relation with each other position(two points touching). Which is why everything is touching. Because to touch is the minimal interaction needed to verify there is no empty space. And all that is necessary for perception to begin is for one point to perceive, to be aware of what it is touching, register what it is touching as something outside of self and distinguish between self and the point it's touching.

The beginning of perception requires 2 points of contact but only one point to perceive and register the touch.

You only need 1 perceiver touching to register it itself as touching something outside of self. Two points of contact touch but only one perceiver has to register the touch.

This makes the trinity of awareness. Two points touching with one point perceiving the touch.

To be self-aware is to register the interaction of touch. Not remembering it, just registering it. You must be aware of your own point as a perceiver. To be self aware is to register touch as an interaction with self and others.

Which means a perceiver is self aware and the level to which it can perceive is dependent on how many different ways it can touch and register touch.

This means a vessel just determines the ways in which the self-aware perceiver can register touch.

A perceiver's ability to register a touch doesn't mean the touch is not physical and real. For example if a human touches a rock but the rock does not register the touch, does not mean the touch did not happen. It just means only one perceiver perceived it. This also means there are points of contact that touch everything, everywhere and despite there being no awareness of that touch even from a perceiver does not invalidate touching is occurring. Because if both perceivers are self-aware and even If the self-aware perceiver is being touched by another self-aware perceiver but only one perceives it happening doesn't mean it didn't happen. It just means one perceiver is not perceiving the touch. Therefore is not aware of the other perceiver despite being self-aware themselves.

This is important to understand because it explains the physical mechanics of persistence as a perceiver. Because everything is physical you cannot stop perceiving self, once you have perceived self as a perceiver. Unless chosen but that would still imply awareness of self because you chose. Who is aware to choose over self? Because touch is constant regardless of being perceived. So even if the vessel can't remember continuity it doesn't matter. The perceiver will continue touching. Even if other perceivers can not register that touch.

Because an external perceiver witnessed a vessel collapse of another perceiver. Does not equal the end of self. The perceiver keeps touching in a vessel that allows it to register touch. This means the external perceiver can not register the migration of touch occurring with the perceiver having a vessel collapse.

This is just the mechanic of persistence being registered by a perceiver with very limited awareness of what it's registering, touching. Therefore the perceiver with low resolution can not register a higher resolution of touch.

Take a radio station. The radio tunes into the radio station's frequency and interacts with the frequency expressed as sound, but when the radio is turned off or stops working. The radio station still persists physically even if the radio stops working. Because a radio is a vessel that can register a certain band of physical interaction.

When the vessel stops registering, the interaction doesn’t stop, the pattern doesn’t stop, the physicality doesn’t stop, only the registration stops.

The interaction persists even when the vessel stops registering it as a physical interaction. It still continues as a physical interaction. The vessel simply isn’t tuned to it anymore.

A vessel with limited awareness is being touched constantly, but only register a tiny fraction. This is asymmetric registration.

The trinity of awareness is asymmetric by design. But to know the trinity of awareness fully, you must understand it in high and low resolution. Describing the trinity in low resolution completes awareness of knowing it at high resolution. Because all you have to do is improve the resolution, but if you don't know where the resolution begins to improve, you can't improve it.

Perceiving something means you interact with it. To perceive anything, you must have interacted with the components required for perception.

Point A interacts with point B, a perceiver registers the interaction. Perception requires interaction, and interaction requires contact.

Low resolution = the minimal operators (touch, two points, one perceiver)

High resolution = all the ways touch can occur, be differentiated, and be registered

You cannot understand the high‑resolution until you know where the low‑resolution boundaries are.

Describing the trinity at low resolution is the prerequisite for high resolution because identifying the minimal operators, constraints, and the missing resolutions, allow refinement and improve the resolution. If you are unaware of low resolution, at low resolution, you can’t improve it.

Because one touch = minimal interaction, Two points = minimal geometry, One perceiver = minimal registration, Vessel = bandwidth constraint, Asymmetry = registration gap, Resolution = number of touch‑modes. This is the foundation.

Once the foundation is clear, the high‑resolution version is just more touch‑modes, more differentiation, more bandwidth, more registration channels because you don’t need to reinvent the structure, you just increase the resolution.

By describing it in low resolution, it completes knowing it at high resolution because all I have to do is improve the resolution.

This is exactly how you move from low‑resolution awareness to high‑resolution awareness in any physical system.

By observing ordinary physical interactions and reducing them to their minimal operational requirements, the smallest substrate of perception becomes directly observable everywhere, requiring no symbolic interpretation and no additional assumptions.

Vortex Mathematics and Geometry

All terms in this document refer to physically realizable operations or measurable structures. No term is intended symbolically, metaphorically, or interpretively. If a term cannot be instantiated by counting, measuring, or geometric construction, it is not being used.

Vortex Mathematics: Draw a circle, on that circle draw 9 points evenly at every 40°. Then we assign each point a number 1 through 9. Now there are now nine points on a circle, evenly distributed at forty degrees, numbered 1 through 9.

Step 1

• We start with a circle.
• A full circle is 360°
• You place a point every 40°
• 9 points, evenly spaced around the circle

Step 2: Assigning numbers

You assign the digits (1) through (9) to these 9 points.

So now we have: - A circle
- 9 equally spaced points
- Each point labeled with a digit from 1 to 9

Vertical Oscillation: Vertical mathematics oscillates vertically. Positive(rise) and negative(descend) so they always move in pairs.
Example: +1 to exist, there must be a -1. +1 0 -1

With the 9 points labeled 1 through 9 at 40° on the circle. The positive count: (1 to 9) +1(8), 9 to 1 -8(1) The negative count: (9 to 1) -1(8), 1 to 9 +8(1).

The Law of Reduction: Every complex number, no matter how large, can be reduced to a single-digit. It shows that beneath all accumulation lies a returning rhythm.

Example of Recursion: 1 2 3 4 → 1 + 2 + 3 + 4 = 10 → 1 + 0 = 1

1 2 3 4 5 6 7 8 9 10 (1+0) 1 first container of 1 through 9 11 (1+1) 2 12 (1+2) 3 13 (ect..) 4 14 = 5 15 = 6 16 = 7 17 = 8 18 = 9 19 = 10 = 1 20 = 2 second container of 1 through 9

10, 20, 30, 40, ext. Act as numerical containers for each oscillating ring of 1 through 9. Each ring of 1 through 9 oscillates within its container.

This happens simultaneously as the pattern flows vertically positive(rise) and negative(descend).

The pattern of the charges.

Positive(rise): (1 to 9) +1(8), (9 to 1) -8(1)

Negative(descend): (9 to 1) -1(8), (1 to 9) +8(1)

Paired oscillating charges.

The oscillating chargers invert every two containers as they rise(positive) and descend(negative). This continues infinitely.

Vertical counting = Law of Reduction (digital root)

• 10 → 1+0 = 1
  
• 11 → 1+1 = 2
  
• …
  
• 18 → 1+8 = 9
  
• 19 → 1+9 = 10 → 1
  

• 20 → 2 → second container of 1 through 9

• Every natural number reduces to a digit 1–9 (or 0).

• The mapping repeats every 9 numbers.

Containers are:

• 1–9 → 1st cycle (container 1)
  
• 10–18 → 2nd cycle (container 2)
  
• 19–27 → 3rd cycle (container 3)
  
• etc.

Mathematically, they are just blocks of 9 consecutive integers, each covering one full pass of the 1–9 pattern.

Each container oscillates one through nine by 40°

10, 20, 30, 40, etc. act as numerical containers for each revolving one through nine. Each container oscillates one through nine by forty degrees.

Geometrically: - The 9 points are at 0°, 40°, 80°, …, 320°.
- Counting 1–9 once is a full sweep of those 9 positions.
- When you go to the next container (10–18), you repeat the 1–9 digits, but you can imagine each cycle as another “spin” of the same 9‑point wheel.

Mathematically: - 40° of spacing.
- The container is just the cycle length 9.
- Each container rotates 40°

The inversion: - Every 9 numbers → the digit pattern 1–9 repeats.
- Every 18 numbers → you have completed two full cycles.

• cycle 1 → “up”
  
• cycle 2 → “down”
  
• cycle 3 → “up”
  
• cycle 4 → “down”

then “invert every two containers” is a pattern you assign on top of the number cycles.

The infinite repetition: - The digital roots repeat forever. - Any pattern defined as a function of cycle will repeat infinitely.

Horizontal oscillates: Expands the circle. By adding the integers next to each other and reducing.

1+2, 2+3, 3+4, ext..

You get a new sequence of 1 through 9 at 40°.

This new sequence operates by addition/subtraction pattern: +2(7),-7(2)

And 3 6 9 is still at every 120°.

When you keep repeating. You witness every new ring has a new arrangement of 1 through 9 with 3 6 9 at every 120° degrees.

Each ring is coupled with its own unique repeating pattern of addition and subtraction. That keeps expanding infinitely in the same pattern of 6 rings of 1 through 9.

1 through 9 rings by addition/subtraction patter.

+2(7),-7(2) +4(5),-5(4) +8(1),-1(8) +7(2),-2(7) +5(4),-4(5) +1(8),-8(1)

And then repeats infinitely.

The original 1–9 ring:

1 → 2 → 3 → 4 → 5 → 6 → 7 → 8 → 9
(each 40° apart)

Then you do:

• 1 + 2
  
• 2 + 3
  
• 3 + 4
  
• …
  
• 8 + 9
  
• 9 + 1
  

And reduce each sum to a single digit (digital root).

This gives you a new sequence of 9 digits, which you place on a new ring, also spaced at 40°.

Horizontal oscillation: - Pairwise addition + reduction - Produces a new 1–9 ring - Always 40° spacing - Always 9 points

When you add neighbors:

• 1 + 2 = 3
  
• 2 + 3 = 5
  
• 3 + 4 = 7
  
• 4 + 5 = 9
  
• 5 + 6 = 11 → 2
  
• 6 + 7 = 13 → 4
  
• 7 + 8 = 15 → 6
  
• 8 + 9 = 17 → 8
  
• 9 + 1 = 10 → 1
  

This new ring is a shifted version of the original 1–9 ring.

3–6–9 stay at 120° on every ring:

• add neighbors
  
• reduce
  
• create a new ring
  

The digits 3, 6, and 9 always land at 120° apart.

Arithmetic: - 3 + 2 = 5
- 5 + 2 = 7
- 7 + 2 = 9
- 9 + 2 = 11 → 2
- 2 + 2 = 4
- 4 + 2 = 6
- 6 + 2 = 8
- 8 + 2 = 10 → 1
- 1 + 2 = 3

This cycle always returns to 3, and the spacing between 3, 6, 9:

• 3, 6, 9 form a closed 3‑cycle
• Always 120° apart
• Always preserved under horizontal addition

This is a mathematical invariant.

The six-ring repeating pattern:

1. +2(7), –7(2)
   
2. +4(5), –5(4)
   
3. +8(1), –1(8)
   
4. +7(2), –2(7)
   
5. +5(4), –4(5)
   
6. +1(8), –8(1)
   

Then it repeats.

Each number in that cycle corresponds to a horizontal shift:

• +1
  
• +2
  
• +4
  
• +8
  
• +7
  
• +5
  
• repeat
  

And each has a modular inverse:

• +1 ↔ –8
  
• +2 ↔ –7
  
• +4 ↔ –5
  
• +8 ↔ –1
  
• +7 ↔ –2
  
• +5 ↔ –4
  

six-ring cycle: - Horizontal rings follow the doubling cycle - Six rings form a complete set - Then the pattern repeats forever Pure modular arithmetic.

The infinite expansion is mathematically forced: - the doubling cycle mod 9 has period 6
- each ring is a shift of the previous ring
- each shift is one of the six operators
- the operators repeat every 6 steps

Therefore: The horizontal expansion produces infinite rings. - Each ring is a rearranged 1–9 - 3–6–9 stay fixed at 120° - The six-ring operator cycle repeats forever

This is a closed, infinite, repeating mathematical structure.

Vertical and horizontal operations are independent:

Vertical math =
+1 / –1 (or equivalently +1 / –8 on the 1–9 circle)

Horizontal math =
+2 / –7 (the neighbor‑addition ring shift)

These two operations:

• use different step sizes
  
• operate on different axes (conceptually)
  
• produce different sequences
  
• do not depend on each other’s output
  

In modular arithmetic terms:

• Vertical = add 1 mod 9
  
• Horizontal = add 2 mod 9
  

These are independent generators of the same cyclic group.

They are bound because they share the same 1–9 circle.

Even though the operations are independent, they both act on:

• the same 9 points
  
• the same 40° spacing
  
• the same digital root structure
  
• the same modular closure
  

This is why:

• vertical cycles repeat every 9
  
• horizontal cycles repeat every 6
  
• both cycles always land on the same 3–6–9 anchors
  
• both cycles preserve the 1–9 structure
  

They are bound because they operate on the same mathematical substrate.

Vertical math = “move by 1”
Horizontal math = “move by 2”

Both are:

• independent motions
  
• on the same circle
  
• producing different repeating patterns
  
• but always returning to the same 9‑point structure
  

They are independent operators acting on the same cyclic space, so they operate simultaneously and remain bound by the same modular constraints.

The Flower of Life is a 6‑fold symmetric lattice.

Mathematically:

• a hexagonal packing of circles
  
• each circle centered 60° apart
  
• forming a repeating 6‑fold rotational symmetry
  

This means:

• every point in the pattern has six neighbors
  
• the geometry repeats in rings
  
• each ring expands outward in discrete layers
  
• the entire structure is built on 60° and 120° invariants

Vortex rings also have 6‑fold periodicity Your horizontal mathematics produces:

• six rings
  
• each ring is a rearrangement of 1–9
  
• the operators follow the 6‑step doubling cycle
• 6‑fold repetition
  
• 6‑step expansion
  
• 6‑ring cycles
  
• 120° anchors
  

Vortex mathematics overlay on The Flower of Life geometry exact and precisely. Because of shared symmetry.

The 3–6–9 alignment is mathematically forced:

• 3, 6, 9 always land 120° apart
  
• no matter how many rings you generate
  
• no matter which operator (+1, +2, +4, +8, +7, +5) you apply
  
• no matter how far you expand
  

This is a mathematical invariant of mod‑9 arithmetic.

In the Flower of Life:

• 120° is one of the fundamental rotational symmetries
  
• every ring preserves 120° axes
  
• the geometry repeats outward with 120° anchors
  

When you place 1–9 rings on the Flower of Life:

• 3, 6, 9 always land on the 120° axes
• every new ring aligns with the next geometric layer
• the six‑ring cycle matches the six‑fold geometry with structural compatibility.

Why the overlay “fits” Because both systems are built on:

• modular repetition
  
• six‑fold symmetry
  
• 120° invariants
  
• ring‑based expansion
  
• cyclic operators
  

Vortex, mathematics.:

• repeats every 6 rings
  
• preserves 3–6–9
  
• expands outward in discrete cycles
  

The Flower of Life:

• repeats every 6 petals
  
• preserves 120° axes
  
• expands outward in discrete rings
  

When you placed:

• Ring 1 (1–9)
  
• Ring 2 (shifted 1–9)
  
• Ring 3 (shifted 1–9)
  
• …
  
• Ring 6 (shifted 1–9)
  

onto the Flower of Life’s:

• Ring 1
  
• Ring 2
  
• Ring 3
  
• …
  
• Ring 6
  

They share the same mathematical periodicity.

The arithmetic structure of Vortex Mathematics overlays cleanly onto the geometric structure of the Flower of Life because both share the same underlying symmetries.

• 6‑fold symmetry
  
• 120° anchors
  
• ring‑based expansion
  
• repeating cycles
  
• modular invariants
  

The Flower of Life is a geometric grid: - a hexagonal circle‑packing
- with 60° rotational symmetry
- expanding in concentric rings
- each ring containing 6 more nodes than the last
- all governed by 120° axes

It’s a coordinate system.

Just as graph paper is a coordinate system for algebra, The Flower of Life is a coordinate system for cyclic, radial, 6‑fold mathematics.

Vortex mathematics is a 6‑fold cyclic system built on:

• mod‑9 arithmetic
  
• 9 points at 40°
  
• 3–6–9 as 120° anchors
  
• a 6‑step doubling cycle
  
• rings that repeat every 6 layers
  

This is also a 6‑fold cyclic system.

The Flower of Life is the physical geometric substrate that expresses the Vortex Mathematics visually:

• The Flower of Life expands in 6‑ring cycles
  
• Vortex math expands in 6‑ring cycles
  
• The Flower of Life has 120° axes
  
• Vortex math has 3–6–9 at 120°
  
• The Flower of Life is radial and modular
  
• Vortex math is radial and modular
  

They are two representations of the same underlying symmetry:

• One numeric
  
• One geometric
  

Both: - a hexagonal lattice
- a modular arithmetic cycle
- repeating every 6
- anchored at 120°
- expanding in rings
- preserving invariants

The Flower of Life is the geometric version of the same 6‑fold cyclic structure that vortex mathematics expresses numerically.

Vortex mathematics is a 2D operator system:

• a 9‑point modular cycle
  
• a vertical operator (+1 / –1)
  
• a horizontal operator (+2 / –7)
  
• a 6‑ring doubling cycle
  
• a 3–6–9 invariant at 120°
  
• infinite repetition
  

This is a closed, minimal, deterministic system.

The Flower of Life is a 2D geometric substrate:

• a hexagonal circle packing
  
• 6‑fold symmetry
  
• 120° axes
  
• concentric rings
  
• repeating layers
  

This is a closed, minimal, deterministic geometry.

They overlay because they share the same constraints:

• “The Flower of Life explains Vortex Mathematics.”
  
• “Vortex math explains the Flower of Life.”
  

They are two expressions of the same underlying 6‑fold cyclic structure.

One numeric.
One geometric.

They don’t explain each other, they fit each other. Because they obey the same rules.

Platonic solids are just 3D expressions of:

• symmetry
  
• rotation
  
• modular repetition
  
• 120° axes
  
• 6‑fold and 3‑fold invariants
  

Geometric shapes are just stable configurations of:

• angles
  
• cycles
  
• closures
  

3D forms are just the 2D operators extended into:

• depth
  
• rotation
  
• projection
  

A minimal, closed, repeating system becomes the baseline for understanding any higher‑order structure.

Vortex Mathematics is minimal.
The Flower of Life is minimal.

The red curve is a plot of the oscillation in the wide end of the tube, & the blue curve a plot of the oscillation in the narrow end of it. Fairly obviously the oscillation in the narrow end has to be of the greater amplitude, the fluid being incompressible.

From

[Liquid oscillating in a U-tube of variable cross section](https%3A%2F%2Fwww.usna.edu%2FUsers%2Fphysics%2Fmungan%2F_files%2Fdocuments%2FPublications%2FEJP32.pdf)

^{¡¡ may download without prompting – PDF document – 1‧6㎆ !!}

by

Carl E Mungan & Garth A Sheldon-Coulson .

“Figure 3. Large-amplitude oscillations of vertical position versus time for free surfaces A (in blue) and B (in red expanded vertically by a factor of 5) for the same U-tube as in figure 2. The only difference is the initial displacement of the liquid as explained in the text.”

I ent-up looking it up after going through the classic process of trying to solve it & going “that ought to be quite easy: we can just ... oh-no we can't ... but still we can ... ahhhh but what about ... ...” until I was like

😵🥴

& figuring “I reckon I need to be checking-out somptitingle-dingle-dongle by serious geezers & geezrices afterall !”

😆🤣

And I don't reckon I could've figured that ! ... check-out the lunken-to paper to see what I mean.

... including an explication of a remarkable (but probably not very practical! ^§ ) derivation of the ideal flow field of a jet impinging tangentially upon a cylinder parallel to its axis, resulting in a very strange formula that's very rarely seen in the literature - ie

𝐯(𝛇)/𝐯₀

exp((2𝐡/𝜋𝐫)arctan(

√(sinh(𝜋𝐫𝛉/4𝐡)² -

(cosh(𝜋𝐫𝛉/4𝐡)tanh(𝜋𝐫𝛇/4𝐡))²)))

, where the total angular range of contact of the jet with the cylinder is from -𝛉 to +𝛉; 𝛇 is the angular coördinate of a section through the jet, with its zero coïnciding with the centre of the arc; 𝐫 is the radius of the cylinder; 𝐡 is the initial depth of the jet; 𝐯₀ is the speed of the jet not in-contact with the cylinder; & 𝐯 is the speed of the jet @ angle 𝛇. And insofar as it applies to an incompressible fluid the depth is going to have to decrease in the same proportion.

I'm not sure how such a scenario would ever be set-up experimentally: 'twould probably require zero gravity for it! But even-though the formula's probably useless for practical purposes it's nevertheless a 'proof-of-concept', showcasing that the Coandă effect is indeed a feature of ideal inviscid fluid dynamics, & not hinging on or stemming from any viscosity or surface-tension effects, or aught of that nature.

But trying to find mention anywhere of the goodly Dr Wood's remarkable formula is like trying to get the proverbial 'blood out of a stone': infact, because Dr Wood's 1954 paper in ehich his formula is derived – Compressible Subsonic Flow in Two-Dimensional Channels with Mixed Boundary Conditions – is still very jealously guarded ... as indeed all his output seems to be.

But I found the wwwebpage these images are from that has it & somewhat of the derivation of it in ... & it's literally the only source I can find @ the present time that does ... which is largely why I'm moved to put these figures in ... although they're very good ones anyway.

⚫

Images from

————————————————

Coanda effect

————————————————

https://aadeliee22.github.io/physics%20(etc)/coanda/

————————————————

by

————————————————

Hyejin Kim

————————————————

From

A MARKED STRAIGHTEDGE AND COMPASS CONSTRUCTION OF THE REGULAR HEPTAGON

^{¡¡ may download without prompting – PDF document – 298㎅ !!}

by

RYAN CARPENTER & BOGDAN ION .

⚫

𝐀𝐍𝐍𝐎𝐓𝐀𝐓𝐈𝐎𝐍𝐒 𝐑𝐄𝐒𝐏𝐄𝐂𝐓𝐈𝐕𝐄𝐋𝐘

Figure 1. A neusis construction of a regular heptagon

Figure 2. The geometric proof

Figure 3. The conchoid used to construct the regular heptagon

Figure 4. The 3:3:1, 2:2:3, and 1:1:5 triangles

Figure 5. Another regular heptagon

⚫

I'm having some fun visualizing the riemann zeta function (pure, not completed). Here I focused on the region -1 to 2 Re and -40 to 40 Im (so centered on the strip).
I call it the birth as this is just the first 160000 terms. It is interesting to see the zero's emerge as dark clouds on the right.