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<p>The objective of this paper is to explore a geometric construction involving successive perpendiculars on the sides of a right-angled triangle to achieve a length of the cube root of 2, thereby solving the Delian problem in Greek mathematics and its connection<i>s to Indian mathematics.</i> Given a right-angled triangle ABC with unit length perpendicular <i>AB</i> and base angle <i>C,</i> perpendiculars are drawn alternately on the hypotenuse AC and base BC. The lengths of these perpendiculars follow a specific pattern: <i>BD/1</i> = <i>DE/BD</i> = <i>EF/DE</i> = <i>FG/EF</i> = <i>GH/FG</i> = <i>HI/GH</i> maintaining a constant ratio equal to the cosine of the base angle <i>C</i>. Consequently, the lengths of successive perpendiculars (<i>BD, DE, EF,</i> etc.) correspond to cosine of angle <i>C</i>, square of cosine of angle <i>C</i>, cube of cosine of angle C… so on respectively. By adjusting angle C while keeping <i>AB</i> fixed at unity, to have results corresponding to the given condition of doubling a cube or a hypercube, as the case may be. This paper presents a dynamic geometric approach, distinct from traditional Euclidean methods, providing a novel perspective on these geometric constructions, particularly in relation to the impossibility of duplicating the cube using only a straightedge and compass.</p>

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Revisiting the delian problem with dynamic geometry: Historical roots and modern innovations

  • 期刊论文

  • 发表日期：

    2026年2月10日

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  • Narinder Kumar Wadhawan

摘要

The objective of this paper is to explore a geometric construction involving successive perpendiculars on the sides of a right-angled triangle to achieve a length of the cube root of 2, thereby solving the Delian problem in Greek mathematics and its connections to Indian mathematics. Given a right-angled triangle ABC with unit length perpendicular AB and base angle C, perpendiculars are drawn alternately on the hypotenuse AC and base BC. The lengths of these perpendiculars follow a specific pattern: BD/1 = DE/BD = EF/DE = FG/EF = GH/FG = HI/GH maintaining a constant ratio equal to the cosine of the base angle C. Consequently, the lengths of successive perpendiculars (BD, DE, EF, etc.) correspond to cosine of angle C, square of cosine of angle C, cube of cosine of angle C… so on respectively. By adjusting angle C while keeping AB fixed at unity, to have results corresponding to the given condition of doubling a cube or a hypercube, as the case may be. This paper presents a dynamic geometric approach, distinct from traditional Euclidean methods, providing a novel perspective on these geometric constructions, particularly in relation to the impossibility of duplicating the cube using only a straightedge and compass.