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”....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.展开更多
This paper presents a graphical procedure for the squaring of a circle of any radius. This procedure, which is based on a novel application of the involute profile, when applied to a circle of arbitrary radius (using ...This paper presents a graphical procedure for the squaring of a circle of any radius. This procedure, which is based on a novel application of the involute profile, when applied to a circle of arbitrary radius (using only an unmarked ruler and a compass), produced a square equal in area to the given circle, which is 50 cm<sup>2</sup>. This result was a clear demonstration that not only is the construction valid for the squaring of a circle of any radius, but it is also capable of achieving absolute results (independent of the number pi (π), in a finite number of steps), when carried out with precision.展开更多
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 ...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.展开更多
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...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.展开更多
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 act...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.展开更多
This paper presents a Method for the squaring of a circle (i.e., constructing a square having an area equal to that of a given circle). The construction, when applied to a given circle having an area of 12.7 cm<sup...This paper presents a Method for the squaring of a circle (i.e., constructing a square having an area equal to that of a given circle). The construction, when applied to a given circle having an area of 12.7 cm<sup>2</sup>, it produced a square having an area of 12.7 cm<sup>2</sup>, using only an unmarked ruler and a compass. This result was a clear demonstration that not only is the construction valid for the squaring of a circle but also for achieving absolute results (independent of the number pi (π) and in a finite number of steps) when carried out with precision.展开更多
文摘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.
文摘This paper presents a graphical procedure for the squaring of a circle of any radius. This procedure, which is based on a novel application of the involute profile, when applied to a circle of arbitrary radius (using only an unmarked ruler and a compass), produced a square equal in area to the given circle, which is 50 cm<sup>2</sup>. This result was a clear demonstration that not only is the construction valid for the squaring of a circle of any radius, but it is also capable of achieving absolute results (independent of the number pi (π), in a finite number of steps), when carried out with precision.
文摘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.
文摘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.
文摘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.
文摘This paper presents a Method for the squaring of a circle (i.e., constructing a square having an area equal to that of a given circle). The construction, when applied to a given circle having an area of 12.7 cm<sup>2</sup>, it produced a square having an area of 12.7 cm<sup>2</sup>, using only an unmarked ruler and a compass. This result was a clear demonstration that not only is the construction valid for the squaring of a circle but also for achieving absolute results (independent of the number pi (π) and in a finite number of steps) when carried out with precision.
