A Procedure for Trisecting an Acute Angle (Method 2)

This paper presents an alternate graphical procedure (Method 2), to that presented in earlier publications entitled, “A Procedure for Trisecting an Acute Angle” and “A Key to Solving the Angle Trisection Problem”. The procedure, when applied to the 30˚ and 60˚ angles that have been “proven” to be nottrisectable and the 45˚ benchmark angle that is known to be trisectable, in each case produced a construction having an identical angular relationship with Archimedes’ Construction, as in Section 2 on THEORY of this paper, where the required trisection angle was found to be one-third of its respective angle (i.e. DE’MA = 1/3 DE’CG). For example, the trisection angle for the 30˚, 45˚ and 60˚ angles were 10.00000˚, 15.00000˚, and 20.00000˚, respectively, and Section 5 on PROOF in this paper. Therefore, based on this identical angular relationship and the numerical results (i.e. to five decimal places), which represent the highest degree of accuracy and precision attainable by The Geometer’s Sketch Pad software, one can only conclude that not only the geometric requirements for arriving at an exact trisection of the 30˚ and 60˚ angle (which have been “proven” to be not-trisectable) have been met, but also, the construction is valid for any arbitrary acute angle, despite theoretical proofs to the contrary by Wantzel, Dudley, and others.

A Simplified Graphical Procedure for Constructing a 10˚or 20˚Angle

This paper presents a simplified graphical procedure for constructing, using an unmarked straightedge and a compass only, a 10˚ to 20˚ angle, which is in other words, trisecting a 30˚ or 60˚ angle. The procedure, when applied to the 30˚ and 60˚ angles that have been “proven” to be not trisectable, produced a construction having the identical angular relationship with Archimedes’ Construction, in which the required trisection angles were found to be 10.00000˚ and 20.00000˚ respectively (i.e. exactly one-third of the given angle or ∠E’MA = 1/3∠E’CG). Based on this identical angular relationship as well as the numerical results obtained, one can only conclude that the geometric requirements for arriving at an exact trisection of the 30˚ or 60˚ angle, and therefore the construction of a 10˚ or 20˚ angle, have been met, notwithstanding the theoretical proofs of Wantzel, Dudley, and others. Thus, the solution to the age-old trisection problem, with respect to these two angles, has been accomplished.

A Key to Solving the Angle Trisection Problem

This paper describes the methodology (or approach) that was key to the solution of the angle trisection problem published earlier in article entitled, “A Procedure For Trisecting An Acute Angle.” It was an approach that required first, designing a working model of a trisector mechanism, second, studying the motion of key elements of the mechanism and third, applying the fundamental principles of kinematics to arrive at the desired results. In presenting these results, since there was no requirement to provide a detailed analysis of the final construction, this information was not included. However, now that the publication is out, it is considered appropriate as well as instructive to explain more fully the mechanism analysis of the trisector in graphical detail, as covered in Section 3 of this paper, that formed the basis of the long sought solution to the age-old Angle Trisection Problem.

The Possibility of Angle Trisection （A Compass-Straightedge Construction） Kimuya M Alex

The objective of this paper is to provide a provable solution of the ancient Greek problem of trisecting an arbitrary angle employing only compass and straightedge （ruler）. （Pierre Laurent Wantzel, 1837） obscurely presented a proof based on ideas from Galois field showing that, the solution of angle trisection corresponds to solution of the cubic equation; x3 - 3x - 1 = 0, which is geometrically irreducible [1]. The focus of this work is to show the possibility to solve the trisection of an angle by correcting some flawed methods meant for general construction of angles, and exemplify why the stated trisection impossible proof is not geometrically valid. The revealed proof is based on a concept from the Archimedes proposition of straightedge construction [2, 3].

Exploring Sums of Interior Angels

This is a lesson integrated with multiple approaches in geometry classroom to deepen middle school students’understanding of geometry and spatial sense in the topic of sums of interior angles in polygons.In three activities,teachers lead students to explore the pattern of interior angles throughout folding paper Origami,constructing animated polygons in Geometer’s Sketchpad,and establishing proof with Parallel Line Theorem.The lesson plan is developed with detailed procedures and prompting questions.The goal of the lesson is to identify the pattern of interior angles in polygons and to analyze the relationship among polygons in the setting of 25 to 30 middle school students.

定向井单弯单稳涡轮钻具造斜率预测

对于定向井单弯单稳涡轮钻具造斜率的预测,用现有的修正三点定圆法预测的是钻具几何曲率,不是井眼曲率,且对造斜能力影响因素考虑不全。利用纵横弯曲法的三弯矩方程,确定井下动力钻具上部切点,结合钻头中心点和下稳定器切点,建立了这三点所对应的井眼轴线上三点坐标,利用三点定圆原理,建立了三点定圆全坐标单弯单稳涡轮钻具造斜率预测模型。实例计算了3种结构涡轮钻具造斜率,对比分析了3种结构涡轮钻具造斜率随涡轮钻具结构角,稳定器与井壁间隙,偏心稳定器偏心距以及稳定器、偏心稳定器、结构角分别到钻头的距离等因素的变化规律,明确了稳定器对单弯涡轮钻具的造斜率影响较小,为减小深井卡钻风险,可不用稳定器或用偏心稳定器替代稳定器安装在近钻头处,能有效提高单弯涡轮钻具的造斜率。该方法综合了纵横弯曲法和三点定圆法的优点,实现了超深井单弯单稳涡轮钻具定向钻井造斜率的简便有效预测。

钻孔刀具缠丝问题的专项改善与研究

为了保证钻削过程的正常进行,必须保证避免出现钻屑过长缠绕刀具的现象,钻屑不能及时被排除,导致孔壁质量无法保证。文章从钻屑的形成过程、刀具几何角度的设计、钻孔参数的优化三方面进行论述,彻底解决钻屑缠绕刀具的问题,为高质量的孔壁状态奠定了充实基础与有效保障。