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This research aims to understand the Hausdorff dimension of Kakeya sets in three-dimensional space using dynamic geometric principles.

July 5, 2026Open Access

Examining Kakeya's n=3 Under Dynamic Euclidean Geometry

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This research aims to understand the Hausdorff dimension of Kakeya sets in three-dimensional space using dynamic geometric principles.
Examined normal translation effects on the dimensional upgrade of line segments to higher dimensions.
Analyzed the implications of extending assumptions from the Perron tree to higher dimensions.
Utilized principles from Dynamic Euclidean Geometry to derive results.
Established that the Hausdorff dimension of a legitimate Kakeya set in ℝ³ equals 3.
Identified that spurious areas can lead to estimates below 3 when incorrect area assumptions are applied (estimated ≈2.5).
Confirmed alignment with Wang–Zahl results that uphold dim_H = 3 under consistent axioms.

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Examines the Hausdorff dimension of the Kakeya set in ℝ³, suggesting implications of dimensional upgrades.

zhigang zhang (2026) studied this question.

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