Notes: Geometric Shapes on a Grid & Figure-of-8 Sequences

Source: Google AI Mode thread, https://share.google/aimode/bO3sC7pZ8NkZlkpxZ (13:53–14:14). Text below is Google AI’s wording; maths shown in readable form. Charts are live: tap a legend item to show/hide it, tap a point for its (x, y).

Geometric Shapes on a Grid

Prompt: insawr geometrical shapes on grid

Geometric shapes can be accurately drawn on a coordinate grid by connecting precise mathematical intersections (lattice points).

Below is an interactive grid visualization featuring a variety of essential 2D geometric shapes, along with a breakdown of their vertex coordinates so you can replicate them perfectly on graph paper.

📐 Vertex Coordinates for Each Shape

If you are drawing these shapes on standard squared grid paper, you can map out their vertices using the following (x, y) coordinate points:

• Square (Blue): Located in the upper left. Vertices at (1, 9), (4, 9), (4, 6), and (1, 6). Every side spans exactly 3 grid units.
• Right-Angled Triangle (Red): Located in the lower left. Vertices at (1, 4), (1, 1), and (4, 1). The right angle sits perfectly on the grid corner at (1, 1).
• Parallelogram (Green): Located in the lower right. Vertices at (6, 1), (9, 1), (10, 4), and (7, 4). The horizontal bases are parallel and exactly 3 units wide.
• Regular Hexagon (Purple): Centered in the upper right. Vertices at approximately (6.5, 7.5), (7, 8.37), (8, 8.37), (8.5, 7.5), (8, 6.63), and (7, 6.63).

Note: While shapes with perpendicular lines (like squares and rectangles) align perfectly with a square grid's lines, regular shapes like equilateral triangles and hexagons require non-integer (decimal) heights due to the mathematical constraints of a square grid.

If you'd like to explore further, let me know if you want to:

• Add more complex polygons like a trapezoid or octagon.
• See how to calculate the area or perimeter of these shapes using the grid units.
• Perform geometric transformations like translating, reflecting, or scaling them across the grid.

Figure-of-8 Number Sequences

Prompt: are there any well known mathematical sequences of numbers that form the figure of 8?

While a standard, single-dimensional sequence of numbers (like 1, 2, 3...) cannot visually form a shape on its own, there are several famous parametric coordinate sequences, knot notations, and topological sequences in mathematics that perfectly define or trace a figure-of-8.

1. Parametric Coordinate Sequences (Lissajous & Lemniscates)

If you generate a sequence of paired numbers (x_n, y_n) using trigonometric functions, they plot a figure-of-8. The most famous is a specific Lissajous sequence, defined by the parametric equations:

By feeding a sequence of regular time steps (t₁, t₂, t₃...) into these formulas, the resulting coordinate sequence loops through the origin and traces a perfect vertical figure-8. A similar sequence generates the lemniscate of Gerono (x_n = cos(t_n), y_n = sin(t_n)cos(t_n)).

2. The Dowker Sequence (Knot Theory)

In topology, knots are classified using mathematical number sequences. The Figure-Eight Knot (the unique prime knot with exactly four crossings) is entirely defined by a short integer sequence known as its Dowker-Thistlethwaite notation:

To get this sequence, mathematicians label the crossings of the figure-8 knot sequentially with alternating odd and even numbers. The Dowker sequence is the specific arrangement of the even numbers paired with the odd ones, mathematically locking the figure-8 shape into a simple string of four digits.

3. Three-Body Problem Choreography Sequence

In physics and celestial mechanics, the three-body problem looks at how three gravitational objects orbit one another. For centuries, it was thought that these orbits were chaotic, but mathematicians discovered a stable "Figure-8 Choreography."

Three equal masses will chase each other infinitely along a single figure-8 path. The sequence of their positions over time is governed by a highly complex sequence of Fourier coefficients, alternating perfectly through 12 distinct geometric configurations (switching between specific straight-line and isosceles triangle alignments).

4. Mayer-Vietoris Sequence (Algebraic Topology)

If you study a "figure-eight space" (topologically known as the wedge sum of two circles), its fundamental structure is analyzed using an algebraic tool called an exact sequence. The Mayer-Vietoris sequence splits the figure-8 into simpler parts to calculate its dimensions and properties:

This specific sequence of abstract mathematical objects yields the algebraic signature of a figure-8.

Would you like to look closer at one of these? I can:

• Provide the exact coordinate points to plot the Lissajous figure-8 yourself.
• Explain how the Dowker sequence numbers {4, 6, 8, 2} are physically derived from the knot crossings.
• Show the animation rules for the three-body figure-8 orbit.

Not plotted: the Dowker sequence, three-body choreography and Mayer-Vietoris sequence. The thread gives text only for these, no coordinates.

Equals Sign in Written Words

Prompt: (=) how is the equation in written word

In a mathematical equation, the symbol "=" is most commonly written out and spoken as "equals" or "is equal to".

Depending on the context, there are a few other ways it is phrased in written word:

• "Is" or "Makes": Often used in simple arithmetic or verbal word problems (e.g., "Two plus two is four" or "Two and three makes five").
• "Yields" or "Results in": Frequently used in applied mathematics, computer science, and chemistry to show a process creating an output (e.g., "The formula yields a value of ten").
• "Is equivalent to": Used in higher mathematics when proving that two different algebraic expressions represent the exact same mathematical object.

If you are looking at a specific formula from earlier (like the figure-8 parametric equations), let me know! I can write out the entire equation in plain, spoken English for you.

Figure-8 on Graph Paper

Prompt: can I see each plotted on the graph paper.

Yes, here is the trigonometric Lissajous sequence plotted directly onto grid graph paper.

To make this easy to plot or visualize on a standard sheet of paper, the math is scaled so that the figure-8 sits perfectly centered at (5, 5) on a 10×10 grid.

📝 Step-by-Step Graph Paper Coordinates

If you want to draw this exact figure-8 on your own graph paper, mark your axis from 0 to 10 on both sides, find the center point at (5, 5), and smoothly connect these sequential coordinate points:

• Start at the center crossing point: (5, 5)
• Loop up and right: Go up to (7.8, 8.5), reach the peak at (5, 9), and round it out through (2.2, 8.5).
• Cross back through the middle: Pass back down exactly through the center (5, 5).
• Loop down and left: Drop down to (7.8, 1.5), reach the bottom base at (5, 1), and round it out through (2.2, 1.5).
• Return home: Head back up to finish right where you started at (5, 5).

Would you like to try plotting a sideways figure-8 (the infinity symbol ∞), or do you want to see the sequence of exact mathematical fractions used to generate these precise grid numbers?

Vertical and Sideways Figure-8 Sequences

Prompt: both please.

Here is how both the vertical and sideways figure-8 loops look when plotted side-by-side, followed by the exact mathematical sequences used to build them.

🧮 The Exact Mathematical Sequence (Fractions of π)

To plot these curves flawlessly, mathematicians use an underlying sequence of angles based on fractions of pi (π). Because trigonometric functions natively scale between −1 and 1, we modify the equations to fit our 10×10 graph paper:

• The Scaling Factor (4): Stretches the loop out so it spans a width/height of 8 grid units.
• The Center Offset (+5): Shifts the entire shape right and up so the crossing point sits at (5, 5) instead of (0, 0).

The table below shows the exact step-by-step numbers generated by the input sequence:

🧭 The Structural Difference in Written Words

Notice how the equations simply swap their frequency multipliers (2t) to turn the shape sideways:

• Vertical Equation: X is equal to five plus four times the sine of t; Y is equal to five plus four times the sine of two t.
• Sideways Equation: X is equal to five plus four times the sine of two t; Y is equal to five plus four times the sine of t.

By doubling the frequency of Y, the vertical shape races up and down twice as fast as it moves left and right, creating stacked loops. Doubling the frequency of X does the reverse, pulling the shape into a wide infinity loop.

Would you like me to generate a printable grid template containing these exact coordinate paths, or would you like to explore how to change the formulas to make the loops thicker, thinner, or asymmetrical?

Print-Ready Coordinates and Morphing the Loops

Prompt: both please.

📄 Print-Ready Coordinate Data (Markdown Text File)

Below is the structured raw data for both curves. You can copy and paste this text directly into a notepad file or spreadsheet to print or keep as a reference guide alongside your graph paper.

========================================================================
FIGURE-8 GRAPH PAPER PLOTTING GUIDE (10x10 GRID)
Center Point of Both Shapes: (5, 5)
========================================================================

--- VERTICAL FIGURE-8 PATH ---
Connect these coordinates in order to trace the upright loop:
1.  (5.00, 5.00)  <- Starting Point (Center Intersection)
2.  (7.83, 9.00)  <- Peak of Upper Right Loop
3.  (9.00, 5.00)  <- Far Right Turning Vertex
4.  (7.83, 1.00)  <- Base of Lower Right Loop
5.  (5.00, 5.00)  <- Midpoint Cross-Through
6.  (2.17, 9.00)  <- Peak of Upper Left Loop
7.  (1.00, 5.00)  <- Far Left Turning Vertex
8.  (2.17, 1.00)  <- Base of Lower Left Loop
9.  (5.00, 5.00)  <- Return Home (Loop Closed)

--- SIDEWAYS INFINITY LOOP PATH ---
Connect these coordinates in order to trace the horizontal loop:
1.  (5.00, 5.00)  <- Starting Point (Center Intersection)
2.  (9.00, 7.83)  <- Peak of the Right-Hand Loop
3.  (5.00, 9.00)  <- Apex Top Edge
4.  (1.00, 7.83)  <- Peak of the Left-Hand Loop
5.  (5.00, 5.00)  <- Midpoint Cross-Through
6.  (9.00, 2.17)  <- Base of the Right-Hand Loop
7.  (5.00, 1.00)  <- Apex Bottom Edge
8.  (1.00, 2.17)  <- Base of the Left-Hand Loop
9.  (5.00, 5.00)  <- Return Home (Loop Closed)

========================================================================

🎨 How to Morph the Loops (Thicker, Thinner, Asymmetrical)

By adjusting specific values in the mathematical equations, you can fully customize how the figure-8 takes shape. We manipulate three parts of the formula: Amplitude (size/width), Phase Shift (symmetry), and Frequency (number of loops).

Let's look at how to alter the standard horizontal infinity equation:

1. Making the Loops Thinner or Thicker

To change the thickness, you adjust the scale factor (amplitude) of the non-doubled variable relative to the other.

• For a Thinner Infinity: Lower the multiplier for y. If you change it to y = 5 + 2 sin(t) while keeping x wide, the loops flatten out vertically like a squashed ribbon.
• For a Thicker/Fatter Infinity: Lower the multiplier for x relative to y. Changing it to x = 5 + 3 sin(2t) and y = 5 + 4 sin(t) makes the loops rounder and bulbous, resembling two touching circles.

2. Creating an Asymmetrical "Teardrop" Loop

If you want one side of the figure-8 to be larger or warped differently than the other, you introduce a Phase Shift (α) inside the sine function of the fast variable:

• Asymmetry Formula: x = 5 + 4 sin(2t + 0.5)
• The Result: Shifting the time alignment causes the loop to compress asymmetrically on one side and flare out on the other, creating a dynamic teardrop or aerodynamic aesthetic rather than a perfectly mirrored infinity sign.

Would you like me to generate the exact coordinate tables for one of these altered shapes (like the asymmetrical teardrop) so you can plot it, or should we look at how to rotate the entire shape by an exact angle (like 45°) across your graph paper?

45-Degree Rotated and Teardrop Loops

Prompt: both please

📐 45-Degree Rotated & Teardrop Asymmetrical Coordinates

Below are the exact mathematical coordinate paths to plot these advanced variations on your 10×10 graph paper.

========================================================================
ADVANCED FIGURE-8 PLOTTING GUIDE (10x10 GRID)
Center Point of Both Shapes: (5, 5)
========================================================================

--- 45-DEGREE ROTATED INFINITY LOOP ---
Traces a diagonal figure-8 tilted perfectly between the grid corners.
Connect these coordinates in order:
1.  (5.00, 5.00)  <- Starting Point (Center Intersection)
2.  (7.83, 7.83)  <- Peak of Upper Right Diagonal Loop
3.  (7.83, 2.17)  <- Tip of Lower Right Diagonal Loop
4.  (2.17, 7.83)  <- Tip of Upper Left Diagonal Loop
5.  (5.00, 5.00)  <- Midpoint Cross-Through
6.  (7.83, 2.17)  <- Lower Right Transition Vertex
7.  (2.17, 2.17)  <- Base of Lower Left Diagonal Loop
8.  (2.17, 7.83)  <- Upper Left Transition Vertex
9.  (5.00, 5.00)  <- Return Home (Loop Closed)

--- ASYMMETRICAL TEARDROP INFINITY LOOP ---
Traces an off-balance loop where the right side flares and the left pinches.
Connect these coordinates in order:
1.  (6.92, 5.00)  <- Starting Point (Off-center Intersection)
2.  (8.79, 7.83)  <- Large Outer Bulb (Top Right Peak)
3.  (3.11, 9.00)  <- High Apex Curve
4.  (1.49, 7.83)  <- Small Outer Bulb (Top Left Peak)
5.  (6.92, 5.00)  <- Midpoint Cross-Through
6.  (8.79, 2.17)  <- Large Outer Bulb (Bottom Right Peak)
7.  (3.11, 1.00)  <- Low Apex Curve
8.  (1.49, 2.17)  <- Small Outer Bulb (Bottom Left Peak)
9.  (6.92, 5.00)  <- Return Home (Loop Closed)

========================================================================

🎨 The Mathematical Visualisation on Grid Paper

Here is how these two distinct manipulations transform the standard infinity symbol layout when drawn out completely.

🧬 The Mechanics Behind the Math

How Rotation Works

To tilt the figure-8 by exactly 45 degrees, we take the standard coordinates and run them through a rotation matrix. This mixes the X and Y values together using the sine and cosine of 45° (which equals √2/2 ≈ 0.707).

• The Spoken Formula: New X becomes five plus the average of the original X and Y curves; New Y becomes five plus the difference between the original X and Y curves.
• The Result: The horizontal loops are pulled cleanly along the diagonals of your graph paper square.

How the Teardrop Shifts

By adding a fixed number (+0.5) inside the sine function for X, we create a phase lead. The horizontal engine of the pen reaches its boundaries earlier than the vertical engine. This means the crossing point shifts away from the physical center (5, 5) over to (6.92, 5), packing the left side closely together while giving the right side room to flare open wide.

Now that we have covered shapes, rotations, and distortions, let me know if you would like to:

• See how this 2D figure-8 can be lifted into 3D space to create an interactive mobius strip or a klein bottle concept.
• Explore how a spirograph tool uses multiple interlocking gear circles to generate continuous figure-8 patterns.

Master Formula Reference Chart

Prompt: can you show all please

To capture the full evolution of these curves, we can combine all six distinct variations into a comprehensive, side-by-side design reference guide.

Here is the complete visual layout of every figure-8 sequence we have explored, mapped out across the grid paper:

🗺️ Master Formula Reference Chart

Every shape on the grid above is generated by altering just four basic structural levers in the parametric equation: Horizontal Width (A_x), Vertical Height (A_y), Horizontal Phase Shift (α), and the Rotation Matrix.

The master template equation used to compute all coordinates is:

🧬 Geometric Transformations: In Written Words

To understand how these shapes mathematically communicate, you can think of them as a dance between two independent rhythmic engines—one pushing left-and-right (X), and one pushing up-and-down (Y):

• When the engines match speeds, they create a simple circle or flat diagonal line.
• When one engine runs exactly twice as fast as the other (2t), it forces the path to cross over its own footprint, locking it into a Figure-of-8.
• When you adjust the multipliers out front, you change the physical bounds of how far the engines can reach, stretching the loop into a thin thread or a fat bead.
• When you introduce a phase shift (like +0.5), you knock the engines out of sync time-wise. One engine turns the corner before the other expects it, warping the perfect balance into an organic teardrop loop.

Now that you can see how all these variations overlay on graph paper, would you like to:

• Take a look at the Spirograph gear mechanics to see how two physically rolling circles generate these exact line paths?
• Transition into 3D spaces to see how this flat figure-8 curve maps onto the surface of a Klein Bottle or Mobius Strip?

Curve 5 (45° Rotated) is drawn from the listed points in the section above. This answer gives only the X half of the rotation formula. The full X and Y pair comes later in the thread (see “The Complete 45° Rotation Formula Pair” below), where the correctly rotated curve is plotted.

Coordinates for the Text-Only Sequences

Prompt (03:34): Can you provide the below please: the Dowker sequence, the three-body figure-eight choreography and the Mayer-Vietoris homology sequence (text-only in the thread), plus the Y half of the 45° rotation formula.

1. The Dowker Sequence Coordinate Paths (2D Knot Diagram Projection)

The notation {4, 6, 8, 2} is a topological code defining how a strand passes over and under itself. To draw this specific mathematical knot on a flat 2D coordinate grid, you can plot these 13 chronological tracking coordinates. Connecting them smoothly with a pen will trace the exact geometry of the Figure-Eight Knot (4₁ knot):

Center Offset: (5, 5) on a standard 10×10 graph grid.

2. Three-Body Figure-Eight Choreography (Orbits & Time Steps)

In celestial mechanics, three equal stellar masses (m₁ = m₂ = m₃ = 1) can chase each other infinitely along a single figure-eight path in a flat plane. Below is the precise numerical sequence spanning one complete orbital period (T ≈ 6.3259).

To see the choreography at any single time step, plot all three bodies simultaneously. For example, at Step 1 (t = 0), place Body 1 at the center, Body 2 on the right wing, and Body 3 on the left wing.

Center Offset: Centered at (5, 5) on a 10×10 canvas, scaled to fit comfortably.

3. The Corrected Mayer-Vietoris Homology Sequence

The algebraic representation of a topological figure-eight space (a wedge sum of two circles, denoted as X = S¹ ∨ S¹) had broken syntax symbols. The clean, mathematically accurate long exact homology sequence is written as:

When computed using integers (ℤ) to find the topological structural properties of the figure-eight, it reduces directly to this clean algebraic chain:

What this means in words: It algebraically proves that a figure-eight space has exactly two independent one-dimensional holes (H₁(X) = ℤ ⊕ ℤ), matching the two visual loops of the figure-eight.

4. The Complete 45° Rotation Formula Pair

To tilt a standard parametric loop by exactly 45 degrees across graph paper, the transformation matrix requires matching X and Y outputs. The full equation pair is:

Given that cos(45°) = sin(45°) = √2/2 ≈ 0.7071, the concrete operational equations centered at (5, 5) are:

Would you like to explore how the masses or gravitational constants are configured to keep the three-body choreography from collapsing, or do you want to start building the vector styles for these paths?

The Vectors

Prompt (03:35): Let’s see the vectors.

Here are the raw mathematical vector paths and programmatic blueprints for your document variables.

These vectors are formatted cleanly in SVG XML path data, making them completely readable and native on iOS Safari/mobile devices without needing any external chart libraries.

1. The Dowker Sequence Vector (<path> Spec)

To plot a authentic topological Figure-Eight Knot Diagram on a 2D viewport, the sequential tracking points must be smoothly curved using cubic Bezier spline commands (C). The path passes cleanly over and under its coordinates, scaling within a standard 500x500 screen layout:

<!-- Figure-Eight Knot Projection Vector -->
<svg viewBox="0 0 500 500" width="100%" xmlns="http://w3.org">
  <path d="M 250,250 
           C 375,425 425,325 250,360 
           C  75,325 125,425 250,250 
           C 375,75  425,175 250,140 
           C  75,175 125,75  250,250 Z" 
        fill="none" 
        stroke="#a855f7" 
        stroke-width="5" 
        stroke-linecap="round" 
        stroke-linejoin="round"/>
</svg>

2. Three-Body Figure-Eight Choreography Vector

In physics, the exact path tracked by three bodies chasing one another on a figure-8 loop is calculated using an analytical Fourier approximation. The fundamental parametric equations mapping the base loop track are:

Here is the exact scaled vector loop path for the orbit track:

<!-- Three-Body Choreography Base Track Vector -->
<svg viewBox="0 0 500 500" width="100%" xmlns="http://w3.org">
  <path d="M 250,250 
           C 390,320 480,250 420,180 
           C 360,110 290,180 250,250 
           C 210,320 140,390 80,320 
           C  20,250 110,180 250,250 Z" 
        fill="none" 
        stroke="#22d3ee" 
        stroke-width="4" 
        stroke-dasharray="6,4"/>
</svg>

Note: To animate the three bodies, place three small circles on this single vector strand spaced exactly 120° (1/3 of the total path distance) apart from each other.

3. The 45° Rotated Coordinate Generation Script (JavaScript Array)

To dynamically transform any standard flat shape sequence into a tilted vector asset on the screen, use this coordinate array computation block. It explicitly packages both the X and Y halves of the rotation matrix equation pair:

// Master 45-Degree Rotation Vector Matrix
const centerX = 250;
const centerY = 250;
const angle = 45 * (Math.PI / 180); // Convert to Radians

function getRotatedVectors(basePoints) {
    return basePoints.map(point => {
        // Step 1: Shift points back to origin
        let dx = point.x - centerX;
        let dy = point.y - centerY;
        
        // Step 2: Apply matching X & Y trigonometric transformations
        let rotatedX = centerX + (dx * Math.cos(angle) - dy * Math.sin(angle));
        let rotatedY = centerY + (dx * Math.sin(angle) + dy * Math.cos(angle));
        
        return { x: rotatedX.toFixed(2), y: rotatedY.toFixed(2) };
    });
}

Would you like to wrap these vectors inside an HTML animation framework so you can watch the three stars orbit live on your screen, or should we port these geometric paths into your main Fo8 v0.3 UI build?

Not plotted: the Mayer-Vietoris sequence (algebra only). The rotation script has no input points of its own, so it isn’t plotted separately; the same formula pair is plotted in the 45° chart above.

Corrected Sequences (Verified)

Not Google AI text. This section was built and checked separately, to correct the errors found in the AI’s Dowker table, knot SVG and three-body table. The AI’s wording above is left unchanged.

Figure-Eight Knot (4₁): Correct 4-Crossing Diagram

Standard parametrisation used: x = (2 + cos 2t)·cos 3t, y = (2 + cos 2t)·sin 3t, z = sin 4t, for t from 0 to 2π. The drawing is the flat (x, y) shadow. At each crossing, z decides which strand goes over, and the under-strand is drawn with a gap.

Check: a numerical search over the whole curve finds exactly 4 crossings. They fall at t = multiples of π/6, and over and under alternate along the strand, as an alternating diagram should.

Dowker (Dowker–Thistlethwaite) check: start at t = 0 and move with increasing t, numbering each crossing as you pass it, 1 to 8. Every crossing gets one odd and one even number. The pairs are 1↔4, 3↔6, 5↔8, 7↔2, so reading the even partners of 1, 3, 5, 7 gives {4, 6, 8, 2}, matching the notation. The order passed is C1 over, C4 under, C2 over, C1 under, C3 over, C2 under, C4 over, C3 under. That alternates correctly.

Three-Body Figure-Eight Choreography (Chenciner–Montgomery, Integrated)

Initial conditions (G = m = 1): positions Body 1 (−0.97000436, 0.24308753), Body 2 (0, 0), Body 3 (0.97000436, −0.24308753). Velocities: Body 1 and Body 3 (0.466203685, 0.43236573), Body 2 (−0.93240737, −0.86473146). Period T = 6.32591398.

Integrated with classical RK4 using 144,000 steps per period (Δt = T/144,000), precomputed and stored here as 720 samples per body. Coordinates are true, unscaled units.

Verification: closing error after one period (positions), 3.9×10⁻⁸ (positions and velocities). Energy drift about 2×10⁻¹⁴. Choreography check, Body 1 at time t equal to Body 2 at t + T/3: within 2.7×10⁻⁸.

Straight-line (Euler) alignments happen at every multiple of T/6, and the middle body rotates 2 → 1 → 3 → 2. At T/2, Body 2 is back in the middle with Bodies 1 and 3 swapped. Isosceles alignments fall halfway between, at odd multiples of T/12. The body at the apex sits on the x-axis at ±1.081, with equal sides 1.658 and a base of 0.691.

What Was Corrected

• Dowker table: the AI’s 13 points give 7 straight-line crossings, none where it claims. Steps 4 and 12 are the same point, and its crossing pairs 1/9 and 8/11 are different points. It is replaced by a true 4-crossing diagram whose labels give {4, 6, 8, 2}.
• Knot SVG: the AI’s Bezier path is an hourglass with one real crossing and no over/under gaps. It is replaced by the 4₁ projection above, with gaps at all four crossings.
• Three-body table: the AI’s positions are off the orbit (unequal x/y scaling), and its “isosceles” triangles have three different sides. They are replaced by integrated positions every T/12.
• Alignment timing: straight lines come every T/6 (not at T/4 and 3T/4), with isosceles alignments halfway between. The AI’s step 7 repeating step 1 at 3T/4 is impossible.
• Centre body at T/2: Body 2, the starting centre body, is back in the middle at T/2, not Body 3 as the AI’s table shows.
• AI Fourier curve: its two main coefficients are close (1.0996 vs 1.1008, 0.3384 vs 0.3388), but the small terms are about 10× too small (true ≈ 0.025 and 0.056). Toggle the overlay to see the difference, up to about 0.06 units.
• 45° rotation: the full formula pair from the later answer (section 9) is correct and is plotted there. The note under the Master Formula Reference Chart has been updated to say so.