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Edit:

Aha, I was looking into cones, and I think I've figured out the ratio. Make a cone where $h=\frac{2{\pi} r}{3}$ That works out to $r^3=\frac{9}{4\pi^2}$ which is aprox 0.611. With a circumference of 3.839 and a lateral length of 1.418 * 2, it gives a total stitching of aprox 6.675

Old:
An octahedron of volume 1 would have an edge length of 1.285. The planar shape you would need two of would be 4 equilateral triangles in a U shape. There would be 6 stiches. 1 on each piece to turn the U shape into a bottomless pyramid (2x), then 4 to put the two pyramids together.

Total stitching: 7.71

I was gonna try a dodecahedron and an icosahedron, but each of those has way too many stitches (20 and 12).

Edit:

Aha, I was looking into cones, and I think I've figured out the ratio. Make a cone where $h=\frac{2{\pi} r}{3}$ That works out to $r^3=\frac{9}{4\pi^2}$ which is aprox 0.611. With a circumference of 3.839 and a lateral length of 1.418 * 2, it gives a total stitching of aprox $6.675$

Old:
An octahedron of volume 1 would have an edge length of 1.285. The planar shape you would need two of would be 4 equilateral triangles in a U shape. There would be 6 stiches. 1 on each piece to turn the U shape into a bottomless pyramid (2x), then 4 to put the two pyramids together.

Total stitching: 7.71

I was gonna try a dodecahedron and an icosahedron, but each of those has way too many stitches (20 and 12).

Edit:

Aha, I was looking into cones, and I think I've figured out the ratio. Make a cone where $h=\frac{2{\pi} r}{3}$ That works out to $r^3=\frac{9}{4\pi^2}$ which is aprox 0.611. With a circumference of 3.839 and a lateral length of 1.418 * 2, it gives a total stitching of aprox 6.398

Old:
An octahedron of volume 1 would have an edge length of 1.285. The planar shape you would need two of would be 4 equilateral triangles in a U shape. There would be 6 stiches. 1 on each piece to turn the U shape into a bottomless pyramid (2x), then 4 to put the two pyramids together.

Total stitching: 7.71

I was gonna try a dodecahedron and an icosahedron, but each of those has way too many stitches (20 and 12).

Edit:

Aha, I was looking into cones, and I think I've figured out the ratio. Make a cone where $h=\frac{2{\pi} r}{3}$ That works out to $r^3=\frac{9}{4\pi^2}$ which is aprox 0.611. With a circumference of 3.839 and a lateral length of 1.418 * 2, it gives a total stitching of aprox 6.675

Old:
An octahedron of volume 1 would have an edge length of 1.285. The planar shape you would need two of would be 4 equilateral triangles in a U shape. There would be 6 stiches. 1 on each piece to turn the U shape into a bottomless pyramid (2x), then 4 to put the two pyramids together.

Total stitching: 7.71

I was gonna try a dodecahedron and an icosahedron, but each of those has way too many stitches (20 and 12).

An octahedron of volume 1 would have an edge length of 1.285. The planar shape you would need two of would be 4 equilateral triangles in a U shape. There would be 6 stiches. 1 on each piece to turn the U shape into a bottomless pyramid (2x), then 4 to put the two pyramids together.

Total stitching: 7.71

I was gonna try a dodecahedron and an icosahedron, but each of those has way too many stitches (20 and 12).

Edit:

Aha, I was looking into cones, and I think I've figured out the ratio. Make a cone where $h=\frac{2{\pi} r}{3}$ That works out to $r^3=\frac{9}{4\pi^2}$ which is aprox 0.611. With a circumference of 3.839 and a lateral length of 1.418 * 2, it gives a total stitching of aprox 6.398

Old:
An octahedron of volume 1 would have an edge length of 1.285. The planar shape you would need two of would be 4 equilateral triangles in a U shape. There would be 6 stiches. 1 on each piece to turn the U shape into a bottomless pyramid (2x), then 4 to put the two pyramids together.

Total stitching: 7.71

I was gonna try a dodecahedron and an icosahedron, but each of those has way too many stitches (20 and 12).