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.

Analysis of secondary flow in shell-side channel of trisection helix heat exchangers

The flow characteristics of shell-side fluid in the tube-and-shell heat exchangers with trisection helical baffles with 35° inclined angles are numerically analyzed. The secondary flow distribution of the fluid in the shell-side channel is focused on. The results on meridian planes indicate that in the shell-side channel, the center part of fluid has an outward tendency because of the centrifugal force, and the peripheral region fluid has an inward tendency under the centripetal force. So in a spiral cycle, the fluid is divided into the upper and lower beams of streamlines, at the same time the Dean vortices are formed near the left baffle, and then the fluid turns to centripetal flow near the right baffle. Finally the two beams of streamlines merge in the main flow. The results of a number of parallel slices between two parallel baffles with the same sector in a swirl cycle also show the existence of the secondary flow and some backward flows at the V-gaps of the adjacent baffles. The secondary flows have a positive effect on mixing fluid by promoting the momentum and mass exchange between fluid particles near the tube wall and in the main stream, and thus they will enhance the heat transfer of the helix heat exchanger.

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].

Fast In-Situ Triaxial Remanent Magnetic Field Measurement for Single-Beam SERF Atomic Magnetometer Based on Trisection Algorithm

We demonstrate a method for quickly and automatically detecting all three components of a remanent magnetic field around a shielded spin-exchange relaxation-free(SERF)atomic magnetometer(AM)using the trisection algorithm(TSA)for zero-field resonance(ZFR).To satisfy the measurement of AMs,a resonance light of the ^(87)Rb D1 line with a spectral width of less than 1 MHz is converted to circular polarization by a linear polarizer and a quarter-wave plate.After the light beam has passed through the alkali metal vapor cell,the residual magnetic field can be measured by searching for triaxial ZFR optical peaks.The TSA stably reduces the measurement time to 2.41 s on average and improves the measurement accuracy,significantly outpacing existing methods.The weighted averages of all measurements with corresponding uncertainties are(–15.437±0.022)nT,(6.062±0.021)nT,and(–14.158±0.052)nT on the x-,y-,and z-axes,respectively.These improvements could facilitate more extremely weak magnetic studies in real time,such as magnetoencephalography(MEG)and magnetocardiography(MCG)measurements.

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.

Effects of Shape and Quantity of Helical Baffle on the Shell-side Heat Transfer and Flow Performance of Heat Exchangers

Shape and quantity of helical baffles have great impact on the shell-side performance of helical baffled heat exchangers （HBHE）. In this work, three physical models of HBHE with baffles of different shape （trisection, quadrant and sextant sector） were investigated. Numerical simulations were performed on HBHE at three helix an- gles （10°, 25° and 40°） by the software ANSYS CFX. Analyses of numerical results indicate that the sextant HBHE shows relatively better fluid flow performance because the leakage flow in the triangle area is evidently reduced and the fluid streamline appears much closer to an ideal spiral flow, while the trisection and quadrant HBHE show more scattered and disordered streamline distributions. The convective heat transfer coefficient and pressure drop in three types of HBHE were presented. Further investigations on the shell side performance with different helical baf- fles were implemented by the field synergy theory. Both theoretical and numerical analyses gave support on the re- lations between helical baffle shape and shell-side performance. This paper may provide useful reference for the selection of baffle shade and auantitv in HBHE.

Comparison of heat transfer performances of helix baffled heat exchangers with different baffle configurations

Numerical simulations were performed on flow and heat transfer performances of heat exchangers having six helical baffles of different baffle shapes and assembly configurations, i.e., two trisection baffle schemes, two quadrant baffle schemes, and two continuous helical baffle schemes. The temperature contour or the pressure contour and velocity contour plots with superimposed velocity vectors on meridian, transverse and unfolded concentric hexagonal slices are presented to obtain a full angular view. For the six helix baffled heat exchangers,the different patterns of the single vortex secondary flow and the shortcut leakage flow were depicted as well as the heat transfer properties were compared. The results show that the optimum scheme among the six configurations is a circumferential overlap trisection helix baffled heat exchanger with a baffle incline angle of 20°(20°TCO) scheme with an anti-shortcut baffle structure, which exhibits the second highest pressure dropΔpo, the highest overall heat transfer coefficient K, shell-side heat transfer coefficient hoand shell-side average comprehensive index ho/Δpo.

Constructing Pandiagonal Snowflake Magic Squares of Odd Order

The constructional methods of pandiagonal snowflake magic squares of orders 4m are established in paper [3]. In this paper, the constructional methods of pandiagonal snowflake magic squares of odd orders n are established with n = 6m+l, 6m+5 and 6m+3, m is an odd positive integer and m is an even positive integer 9|6m + 3. It is seen that the number sets for constructing pandiagonal snowflake magic squares can be extended to the matrices with symmetric partial difference in each direction for orders 6m + 1 , 6m + 5; to the trisection matrices with symmetric partial difference in each direction for order 6m + 3.

A Procedure for Trisecting an Acute Angle

This paper presents a graphical procedure, using an unmarked straightedge and compass only, for trisecting an arbitrary acute angle. The procedure, when applied to the 30˚angle that has been “proven” to be not trisectable, produced a construction having the identical angular relationship with Archimedes’ Construction, in which the required trisection angle was found to be exactly one-third of the given angle (or ∠E'MA = 1/3∠E'CG = 10˚), as shown in Figure 1(D) and Figure 1(E) and Section 4 PROOF in this paper. Hence, based on this identical angular relationship between the construction presented and Archimedes’ Construction, one can only conclude that geometric requirements for arriving at an exact trisection have been met, notwithstanding the theoretical proofs of Wantzel, Dudley, and others.

论"二分法"的终结与"三分法"的勃兴

二点论比一点论具有更深刻更果断的特点。但由于它难以在动态中把握持久平衡,无法捕捉两极之间的那个"第三者",因而不能最终战胜形而上学。能够驾驭两极,游刃于"第三者"的只有三点论,它的最初表现形态是由分到合,即由一分为二到合二而一的,一分为三则是它发展及转化的最高环节。所谓一分为三,就是对多样性世界的三分看法,它反映了事物的全面(正面、反面,侧面),把握了矛盾的全体(静态矛盾、动态矛盾,矛盾中间),概括了人类文明的全局(物质文明,政治文明、精神文明),是科学的世界现与方法论。"三"便是万物的基元。

Multi Parameters Golden Ratio and Some Applications

The present paper is devoted to the generalized multi parameters golden ratio. Variety of features like two-dimensional continued fractions, and conjectures on geometrical properties concerning to this subject are also presented. Wider generalization of Binet, Pell and Gazale formulas and wider generalizations of symmetric hyperbolic Fibonacci and Lucas functions presented by Stakhov and Rozin are also achieved. Geometrical applications such as applications in angle trisection and easy drawing of every regular polygons are developed. As a special case, some famous identities like Cassini’s, Askey’s are derived and presented, and also a new class of multi parameters hyperbolic functions and their properties are introduced, finally a generalized Q-matrix called Gn-matrix of order n being a generating matrix for the generalized Fibonacci numbers of order n and its inverse are created. The corresponding code matrix will prevent the attack to the data based on previous matrix.

Determining reference conditions for TN,TP,SD and Chl-a in eastern plain ecoregion lakes,China

Establishing the nutrient reference condition（baseline environmental condition） of lakes in an ecoregion is a critical consideration in the development of scientifically defensible aquatic nutrient criteria.Three methods were applied to determine reference conditions in the Eastern plain ecoregion lakes with respect to total phosphorus（TP）,total nitrogen（TN）,planktonic chlorophyll a（Chl-a） and Secchi depth（SD）.The reference condition value for the lakes in the Eastern plain ecoregion by the trisection method is TP of 0.029 mg/L,TN of 0.67 mg/L,Chl-a of 3.92 mg/m 3,SD of 0.85 m,and the reference condition range by the lake population distribution approach is TP of 0.014-0.043 mg/L,TN of 0.360-0.785 mg/L,Chl-a of 1.78-4.73 mg/m 3,SD of 0.68-1.21 m.Additionally,empirical models were developed for estimating the reference Chl-a concentration and SD successfully for lakes in the Eastern plain ecoregion.Overall,the data suggest that multiple methods can be used to determine reference conditions and that in Eastern plain ecoregion lakes the reference condition corresponds to a mesotrophic status.