purlJam

# Non-Euclidean Poliwag

This pattern creates a unique Poliwag featuring a **hyperbolic tail**. In Euclidean geometry, the circumference of a circle is $2\pi r$. In the non-Euclidean geometry of this tail, we increase stitches so rapidly that the surface area exceeds what can lay flat, forcing the fabric to ruffle into beautiful, chaotic folds.

**Difficulty**: Beginner
**Size**: Approximately 4 inches (10 cm) tall, excluding tail.

### Materials
*   **Yarn**: Worsted weight (Size 4)
    *   Color A: Bright Blue (approx. 80 yards)
    *   Color B: White (approx. 10 yards or white felt)
    *   Color C: Black (scraps for spiral/eyes)
*   **Needles**: US 5 / 3.75 mm Double Pointed Needles (DPNs) or a long circular needle for Magic Loop.
*   **Notions**: Polyester fiberfill, tapestry needle, 10mm safety eyes (optional).

### Specifications & Gauge
*   **Gauge**: 20 sts and 26 rows = 4 inches (10 cm) in stockinette stitch. 
*   *Note: Gauge is not critical, but the fabric must be tight enough so stuffing does not peek through.*

### Abbreviations (US)
*   **k**: knit
*   **k2tog**: knit 2 stitches together (decrease)
*   **m1**: make 1 stitch (increase)
*   **st(s)**: stitch(es)
*   **rnd(s)**: round(s)
*   **[ ]**: repeat instructions within brackets
*   **( )**: total stitch count at the end of the round

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## Instructions

### 1. The Body (Spherical Geometry)
Using **Color A**, cast on 6 sts. Join for working in the round.
*   **Rnd 1**: [k1, m1] around. ($6 + 6 = 12$ sts)
*   **Rnd 2**: [k1, m1] around. ($12 + 12 = 24$ sts)
*   **Rnd 3**: [k2, m1] around. ($24 + 12 = 36$ sts)
*   **Rnd 4**: [k5, m1] around. ($36 + 6 = 42$ sts)
*   **Rnds 5–15**: Knit all sts. (11 rounds total)
*   **Rnd 16**: [k5, k2tog] around. ($42 - 6 = 36$ sts)
*   **Rnd 17**: [k4, k2tog] around. ($36 - 6 = 30$ sts)
*   **Rnd 18**: [k3, k2tog] around. ($30 - 6 = 24$ sts)
*   *Begin stuffing the body firmly with fiberfill.*
*   **Rnd 19**: [k2, k2tog] around. ($24 - 6 = 18$ sts)
*   **Rnd 20**: [k1, k2tog] around. ($18 - 6 = 12$ sts)
*   **Rnd 21**: [k2tog] around. ($12 - 6 = 6$ sts)
Cut yarn, leaving a 10-inch tail. Thread through remaining stitches and pull tight to close.

### 2. The Non-Euclidean Tail (Hyperbolic Plane)
Unlike a flat tail, this tail uses a constant growth rate to create negative curvature. 
Using **Color A**, cast on 10 sts. We will work this flat (back and forth).
*   **Row 1 (RS)**: Knit all sts. (10 sts)
*   **Row 2 (WS)**: Purl all sts. (10 sts)
*   **Row 3**: [k1, m1] in every stitch. ($10 \times 2 = 20$ sts)
*   **Row 4**: Purl all sts. (20 sts)
*   **Row 5**: [k1, m1] in every stitch. ($20 \times 2 = 40$ sts)
*   **Row 6**: Purl all sts. (40 sts)
*   **Row 7**: [k1, m1] in every stitch. ($40 \times 2 = 80$ sts)
*   **Row 8**: Purl all sts. (80 sts)
*   **Row 9 (Bind off row)**: Bind off all stitches loosely.
*   *Result: You will have a highly ruffled, "brain-like" ruffle that cannot be flattened against a table.*

### 3. Belly Disk (Optional)
*Note: You can use a circle of white felt for a beginner-friendly option, or knit this disk.*
Using **Color B**, cast on 6 sts.
*   **Rnd 1**: [m1] in every st. (12 sts)
*   **Rnd 2**: [k1, m1] around. (18 sts)
*   **Rnd 3**: [k2, m1] around. (24 sts)
Bind off and weave in ends.

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## Assembly

1.  **The Belly**: Center the White Belly Disk (or felt circle) on the front of the body. Sew in place using a whip stitch. 
2.  **The Spiral**: Using **Color C** and a tapestry needle, embroider a spiral starting from the center of the white belly, moving outward.
3.  **The Face**: 
    *   Place safety eyes or embroider two black circles above the belly. 
    *   Embroider a small pink or red mouth just below the eyes.
4.  **The Tail**: 
    *   Take the "cast-on" edge of the Tail (the 10-stitch edge). 
    *   Sew this edge to the back of the Poliwag body. 
    *   Because the tail is hyperbolic, it will naturally ruffle outward. You do not need to style it; its own geometry prevents it from being flat!

**Designer Note**: To see the "Non-Euclidean" effect, try to press the tail flat against a table. You will notice it is impossible without the fabric overlapping—this is the physical manifestation of negative curvature!