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Given a rectifiable curve C : [0, 1] → R² in the plane, it makes sense to roll a unit sphere S² on the plane along that curve and ask what is its net rotation in SO(3).

I wonder if this also makes sense for a wider class of curves than just the rectifiable ones.

For simplicity let C : [0, 1] → R² be a homeomorphism onto its image C([0, 1]), and assume C([0, 1]) has constant Hausdorff dimension d with 1 < d < ∞. (I.e., every neighborhood of every point of C([0, 1]) has the same Hausdorff dimension d.)

Is it possible that, although C([0, 1]) is not rectifiable, it nevertheless makes sense to ask what the net rotation is of a unit sphere rolled along it from beginning to end?

For definiteness, we can consider the "Koch curve" (2nd paragraph of https://mathworld.wolfram.com/KochSnowflake.html) that is one-third of the Koch snowflake curve.

Call this Koch curve W. W is the limit of a sequence of standard stages W_n as n → ∞.

Its Hausdorff dimension is easily computed to be log₃4 = 1.2618595+. Since this exceeds 1, it cannot be rectifiable.

If net rotation makes sense for this curve, what element of SO(3) is it? (Assume that Stage 1, the "witch's hat" of four line segments, has euclidean length = 4/3.)

I would be happy to define this concept as the limit as n → ∞ of the net rotation of S² along the nth stage W_n if that limit exists.

Given a rectifiable curve C : [0, 1] → R² in the plane, it makes sense to roll a unit sphere S² on the plane along that curve and ask what is its net rotation in SO(3).

I wonder if this also makes sense for a wider class of curves than just the rectifiable ones.

For simplicity let C : [0, 1] → R² be a homeomorphism onto its image C([0, 1]), and assume C([0, 1]) has constant Hausdorff dimension d with 1 < d < ∞. (I.e., every neighborhood of every point of C([0, 1]) has the same Hausdorff dimension d.)

Is it possible that, although C([0, 1]) is not rectifiable, it nevertheless makes sense to ask what the net rotation is of a unit sphere rolled along it from beginning to end?

For definiteness, we can consider the "Koch curve" (2nd paragraph of https://mathworld.wolfram.com/KochSnowflake.html) that is one-third of the Koch snowflake curve.

Call this Koch curve W. W is the limit of a sequence of standard stages W_n as n → ∞.

Its Hausdorff dimension is easily computed to be log₃4 = 1.2618595+. Since this exceeds 1, it cannot be rectifiable.

If net rotation makes sense for this curve, what element of SO(3) is it? (Assume that Stage 1, the "witch's hat" of four equal line segments, has euclidean length = 4/3.)

I would be happy to define this concept as the limit as n → ∞ of the net rotation of S² along the nth stage W_n if that limit exists.

Given a rectifiable curve C : [0, 1] → R² in the plane, it makes sense to roll a unit sphere S² on the plane along that curve and ask what is its net rotation in SO(3).

I wonder if this also makes sense for a wider class of curves than just the rectifiable ones.

For simplicity let C : [0, 1] → R² be a homeomorphism onto its image C([0, 1]), and assume C([0, 1]) has constant Hausdorff dimension d with 1 < d < ∞. (I.e., every neighborhood of every point of C([0, 1]) has the same Hausdorff dimension d.)

Is it possible that, although C([0, 1]) is not rectifiable, it nevertheless makes sense to ask what the net rotation is of a unit sphere rolled along it from beginning to end?

For definiteness, we can consider the "Koch curve" (2nd paragraph of https://mathworld.wolfram.com/KochSnowflake.html) that is one-third of the Koch snowflake curve.

Call this Koch curve W. W is the limit of a sequence of standard stages W_n as n → ∞.

If net rotation makes sense for this curve, what element of SO(3) is it? (Assume that Stage 1, the "witch's hat" of four line segments, has euclidean length = 4/3.)

I would be happy to define this concept as the limit as n → ∞ of the net rotation of S² along the nth stage W_n if that limit exists.

Given a rectifiable curve C : [0, 1] → R² in the plane, it makes sense to roll a unit sphere S² on the plane along that curve and ask what is its net rotation in SO(3).

I wonder if this also makes sense for a wider class of curves than just the rectifiable ones.

For simplicity let C : [0, 1] → R² be a homeomorphism onto its image C([0, 1]), and assume C([0, 1]) has constant Hausdorff dimension d with 1 < d < ∞. (I.e., every neighborhood of every point of C([0, 1]) has the same Hausdorff dimension d.)

Is it possible that, although C([0, 1]) is not rectifiable, it nevertheless makes sense to ask what the net rotation is of a unit sphere rolled along it from beginning to end?

For definiteness, we can consider the "Koch curve" (2nd paragraph of https://mathworld.wolfram.com/KochSnowflake.html) that is one-third of the Koch snowflake curve.

Call this Koch curve W. W is the limit of a sequence of standard stages W_n as n → ∞.

Its Hausdorff dimension is easily computed to be log₃4 = 1.2618595+. Since this exceeds 1, it cannot be rectifiable.

If net rotation makes sense for this curve, what element of SO(3) is it? (Assume that Stage 1, the "witch's hat" of four line segments, has euclidean length = 4/3.)

I would be happy to define this concept as the limit as n → ∞ of the net rotation of S² along the nth stage W_n if that limit exists.

Given a rectifiable curve C : [0, 1] → R² in the plane, it makes sense to roll a unit sphere S² along that curve and ask what is its net rotation in SO(3).

I wonder if this also makes sense for a wider class of curves than just the rectifiable ones.

For simplicity let C : [0, 1] → R² be a homeomorphism onto its image C([0, 1]), and assume C([0, 1]) has constant Hausdorff dimension d with 1 < d < ∞. (I.e., every neighborhood of every point of C([0, 1]) has the same Hausdorff dimension d.)

Is it possible that, although C([0, 1]) is not rectifiable, it nevertheless makes sense to ask what the net rotation is of a unit sphere rolled along it from beginning to end?

For definiteness, we can consider the "Koch curve" (2nd paragraph of https://mathworld.wolfram.com/KochSnowflake.html) that is one-third of the Koch snowflake curve.

Call this Koch curve W. W is the limit of a sequence of standard stages W_n as n → ∞.

If net rotation makes sense for this curve, what element of SO(3) is it? (Assume that Stage 1, the "witch's hat" of four line segments, has euclidean length = 4/3.)

(I would be happy to define this concept as the limit of the net rotation of S² along the nth stage W_n as n → ∞, if that limit exists.)

Given a rectifiable curve C : [0, 1] → R² in the plane, it makes sense to roll a unit sphere S² on the plane along that curve and ask what is its net rotation in SO(3).

I wonder if this also makes sense for a wider class of curves than just the rectifiable ones.

For simplicity let C : [0, 1] → R² be a homeomorphism onto its image C([0, 1]), and assume C([0, 1]) has constant Hausdorff dimension d with 1 < d < ∞. (I.e., every neighborhood of every point of C([0, 1]) has the same Hausdorff dimension d.)

Is it possible that, although C([0, 1]) is not rectifiable, it nevertheless makes sense to ask what the net rotation is of a unit sphere rolled along it from beginning to end?

For definiteness, we can consider the "Koch curve" (2nd paragraph of https://mathworld.wolfram.com/KochSnowflake.html) that is one-third of the Koch snowflake curve.

Call this Koch curve W. W is the limit of a sequence of standard stages W_n as n → ∞.

If net rotation makes sense for this curve, what element of SO(3) is it? (Assume that Stage 1, the "witch's hat" of four line segments, has euclidean length = 4/3.)

I would be happy to define this concept as the limit as n → ∞ of the net rotation of S² along the nth stage W_n if that limit exists.