Logarithmic Spiral Embroidery

A logarithmic spiral or log spiral is a self-similar curve that grows outward but maintains the same shape, making a constant angle with the radial lines from its center. Log spirals grow, which is why they are commonly found in nature. All living things must grow. Logarithmic spiral embroidery uses the self-similar to create patterns, stitched by spiraling outward. By doing this project, students will learn about the relationship between math, computation, and creativity.

This project combines coordinate geometry, block-based coding, generative AI and art/design to create a logarithmic spiral embroidery design (see image).

Coordinate geometry is a branch of math that uses a coordinate system (like the x-y plane) to study shapes and positions, translating geometric ideas into algebraic equations and vice-versa, allowing us to use algebra to solve geometry problems.

Block-based coding is a visual way to program by dragging and snapping colorful puzzle-piece-like blocks, each representing a command (ex. move or draw), to build sequences of instructions without typing syntax, making it easy for people to learn coding logic and create designs.

Embroidery is the art or process of forming decorative designs with hand or machine needlework. One of the biggest trends in contemporary embroidery is the rise of geometric patterns and motifs. For this project, you will create an embroidery using colorful logarithmic spirals that grow outward from the center (small stitches or wefts get bigger).

For math/coding:

Go to the CSDT website and navigate to Graffiti Grapher (to make a logarithmic spiral), or

Use the warping template (see step 2)

Optional: Use a protractor, ruler, compass, pencil, paper (to practice drawing log spirals)

For the embroidery project:

Use a circle embroidery hoop with a tightening screw

Spool of strong string or warp thread

Wool roving (also felting wool) in assorted colors

Tapestry needles with large eyes (large eye sewing needles)

Needle threaders

Pencil or marker for marking on inner hoop

Scissors, when necessary

Warping template (see step 3)

Markers in assorted colors

Optional: Embroidery thread in assorted colors

Use a Culturally Situated Design Tool or CSDT to practice making a logarithmic spiral.

CSDTs are a set of Java-based and Snap! tools that were created to engage learners in math and computer science by connecting concepts to diverse cultural contexts. For this project, use Graffiti Grapher to code a logarithmic spiral.

1. Launch the Graffiti Grapher CSDT
2. Navigate to "Software"
3. In the Script Area, remove the blocks under the "set pen size to (12)"
4. Click and drag the "log spiral" Pen blocks from the Blocks Palette
5. Click the green area to run the code on the Stage (make additions/revisions as needed)
6. Screen save and print the finished design

Examples (pics) are provided in this step but it is recommended that learners create the log spiral without additional help.

You can draw a logarithmic spiral using a protractor, ruler, and compass by plotting points whose distance from the center increases by a constant factor as the angle increases. Here are the steps:

1. Mark the center (point 0) that will be the origin of the log spiral.
2. Use a ruler to draw draw a starting radius (straight line from 0)
3. Divide the circle into equal angles
4. Choose a growth factor
5. A logarithmic spiral grows by a constant ratio each step.
6. Example: Multiply distance by 1.2 each step or 1.25 each step
7. Measure spiral points
8. Draw the spiral curve
9. Lightly connect the points with a smooth continuous curve.
10. Extend the spiral
11. Continue the process:
12. Add more angle divisions
13. Keep multiplying the radius by the same factor
14. Plot and connect new points
15. The curve will expand outward in a logarithmic spiral.

Here's a video by Arthur Geometry explaining the process: https://www.youtube.com/watch?v=qxM39DKAm6w

Creating a design for an embroidery involves selecting a motif (ex. log spiral), choosing appropriate stitch types (ex. soumak, back, running, couching), and planning the color palette. For this step, print out and use the attached "warping template" to color in your log spiral design (see image).

You can use an assortment of colored markers, pencils or even crayons to color in your design using the warping template (see image).

You can also use AI image generators such as ChatGPT or Midjourney to create a design (see image).

For this project, the log spiral does not have to be perfect. The purpose of this step is to show a relationship between the log spiral, coordinate geometry and embroidery (art/design).

This step shows how to warp an embroidery hoop for creating the log spiral embroidery.

Warp refers to strong, lengthwise threads that are held under tension on a hoop, forming the stable foundation ("verticals") onto which the weft (crosswise threads) is woven over and under to create the project or design.

A warping template (PDF) is attached a handy guide to ensure the warp threads are evenly distributed across the embroidery hoop. If you can't print the template:

1. Trace the inner hoop on a piece of paper.
2. Fold your paper double two times. The centre of your hoop should now be one of the corners.
3. Fold your paper such that you create a triangle.
4. Double (or triple) your triangle.
5. Now open up the folded paper and make a mark where each fold line meets the drawn embroidery hoop.
6. Use a pencil to transfer the stripes from the warping drawing or template to the outside of the inner embroidery hoop.

Next, wrap the warp thread around the inner hoop:

1. Tie one end of the thread with a single knot covering one if the pencil marks (see example).
2. Pull the warp thread straight across to the other side of your hoop. Bring the thread to the backside of your embroidery hoop and wrap it around to the front. Note: Use a empty spool (or another small object) to wrap the threads.
3. Repeat the process, wrapping thread around each pencil mark on the inner hoop.
4. When you wrap the last thread (before the first knot), pull the spool and thread down and wrap it around the center to bring the center threads tighter together.
5. Bring the string to the first knot, tie a second knot and cut the excess off.
6. Re-insert the inner hoop in the larger one and tighten the screw.

Almost every weaving or embroidery has warp threads, which are the vertical, structural foundation threads set up on a loom or frame, running parallel to the horizontal or weft threads.

You can use the soumak weaving technique (see chart) to create a slightly raised, embroidered-looking surface by wrapping colored weft threads or strips of wool roving around the warp threads.

Soumak is an ancient, durable flat-weaving technique originating from the Caucasus region—likely around Shemaka in the 17th century—used to create decorative textiles, rugs, and tribal bags. It involves wrapping colored weft threads over and under warps, creating a smooth, embroidered front and a rugged, textured back.

Key features of soumak weaving include:

1. Raised, rope-like pattern lines
2. Clear geometric motifs
3. A smoother front surface and loose threads on the back

Because the threads wrap around multiple warps, the result looks almost embroidered onto the surface, even though it is part of the weave structure.

For tiny wefts: Use embroidery thread and/or a thin tuft of felting wool threaded through a needle. Use a needle threader to get the material through a large needle eye. Use the needle to wrap thread/wool around single warp threads (see image).

For bigger wefts: Use thicker tufts of wool roving and weave them in and out of the warp threads. Use your fingers to wrap wool around double warp threads (see image).

The Weaving Loom has another example of this technique: https://www.theweavingloom.com/weaving-techniques-circular-soumak-finish

When finished insert the smaller frame into the larger hoop and tighten it. Display it as you wish.