(p, q) Torus Knot

A parametric torus knot curve winding over a glass torus

Category: MathematicalLevel: Beginner30 components · 34 wires

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A (p, q) torus knot winds p times around the axis of symmetry and q times through the hole of the torus. Its points are computed with x = (R + r cos(q t)) cos(p t), y = (R + r cos(q t)) sin(p t), z = r sin(q t), then interpolated periodically and piped. The reference torus is a Revolution of a circle about the Z axis and is shown as glass. When p and q are coprime the curve is a single knot; otherwise the winding repeats over a shorter loop.

Grasshopper definition with wires

The real Grasshopper canvas: 30 components joined by 34 wires.

(p, q) Torus Knot: Grasshopper canvas with components and wires

The result in Rhino

Project info

CategoryMathematical
LevelBeginner
ToolGrasshopper · Rhino 8
Price (one price)US$123
Components30
Wires34

The project file you can download

The real project file is available. It is not just the pictures on this page: you get the working definition with every component and wire in place. It can be downloaded after payment, and we send the download link by WhatsApp or email.

ZIP: .gh + .ghx + baked .3dm + images

What is inside the file

The original Grasshopper definition (.gh and .ghx) with every component and wire in place, ready to open and tweak
The geometry baked to Rhino (.3dm)
Canvas and result images
A BACA-DULU.txt file listing the plugins required, the version and how to use it
All packed in one ZIP file

Project file: The actual project file (not just the picture) is available to download once payment is complete; we send the download link by WhatsApp or email
Payment: One price per project, paid in full upfront; the file is sent once payment is received
Tax: Prices exclude 11% VAT
Customisation: Adapting it to your own size or design can be ordered separately via WhatsApp

Components and plugins

Components used

Range
Interpolate
Pipe
Circle
Line SDL
Revolution

Plugins required

None. It uses only the components built into Grasshopper.

Parameters you can change

p (1 to 12): windings around the axis
q (1 to 12): windings through the hole
Major Radius (3 to 20): torus radius

How the definition works

1Set the sliders in the Parameters group (p, q, Major Radius).
2Angles: Range t, then p t and q t with Multiplication.
3Knot equations: Cosines and sines of p t and q t are combined by the torus knot formula.
4Pipe & torus: The curve is piped; the reference torus is a Revolution of an XZ circle about the Z axis.

Price and ordering

US$123

One price for the whole project file, excl. 11% VAT

Order the file via WhatsApp

Frequently asked questions

Do I get the real file?

Yes. The pictures on this page are only previews. After payment we send the download link of the real project file (a ZIP containing the .gh and .ghx definition, the baked .3dm, images and a BACA-DULU.txt note) by WhatsApp or email.

Which Grasshopper version do I need?

The file was made in Rhino 8 with Grasshopper (Rhino 8). Newer versions open it too.

Can you adapt it to my design?

Yes, as a separate order. Message us on WhatsApp with a short description of the size and shape you want.

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Images are real captures of Rhino 8 and Grasshopper taken from the project files. Rhino and Grasshopper are trademarks of Robert McNeel & Associates.

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The project file you can download

ZIP: .gh + .ghx + baked .3dm + images

What is inside the file

The original Grasshopper definition (.gh and .ghx) with every component and wire in place, ready to open and tweak

The geometry baked to Rhino (.3dm)

Canvas and result images

A BACA-DULU.txt file listing the plugins required, the version and how to use it

All packed in one ZIP file

Project file: The actual project file (not just the picture) is available to download once payment is complete; we send the download link by WhatsApp or email

Payment: One price per project, paid in full upfront; the file is sent once payment is received

Tax: Prices exclude 11% VAT

Customisation: Adapting it to your own size or design can be ordered separately via WhatsApp

How the definition works

1Set the sliders in the Parameters group (p, q, Major Radius).

2Angles: Range t, then p t and q t with Multiplication.

3Knot equations: Cosines and sines of p t and q t are combined by the torus knot formula.

4Pipe & torus: The curve is piped; the reference torus is a Revolution of an XZ circle about the Z axis.