直角三角形生成關係的研究與發展
k(2αβ ,α2 ? β2,α2 + β2 )是大家熟悉畢氏定理的通式解，且一般書籍的証明大都採用代數的手法證明。以國中生而言，上述的代數方對國中生來說不夠直接且較無推展的實用性。因此幾何觀點出發發展另一種思考方式，利用角平線的性質給予畢氏定理比例解另一種全新的詮釋，並賦予比例解中的參數α 、β 在幾何的意義。在推理的過程中，我們得到一個相當有用的對應關係：一個有理數對應到一個直角三角形、兩個有理數對應到海倫三角形，再將此對應關係運用到各種幾何圖形上面，即可證明出他們所對應的通式解。最後我的興趣鎖定在海倫三角形、完美海倫多邊形與超完美海倫多邊形上的做圖方法上，善用我們所發展的對應關係，上述的問題皆可迎刃而解。k(2αβ ,α2 ? β2,α2 + β2 ) is a popular formula in Pythagoras Theory, often proved in algebra approach among books. Nevertheless, in light of junior high students, the aforementioned algebra method is neither direct nor practical. Hence, a different thinking method is derived from geometry perspective, using the straight line concept to reinterpret Pythagoras Theory and define the geometric meanings of α andβ . In the process of logical development, a useful correlation emerges: a rational number correlates with a straight-angled triangle, and two rational numbers correlate with Heron Triangle. This correlation can be applied to all kinds of geometrical diagrams to prove the correlated homogenous solution. Ultimately, my interest lies in the diagram methods of Heron Triangle, Perfect Heron Polygon, and Super Perfect Heron Polygon in order to apply our developed correlations to solve the above mentioned problems.
形?與形外
在這篇研究報告中,我用了三種觀點來推廣幾何中的反演變換,首先,把反演變換視為是一種圓內與圓外的一種1-1且onto的映射,第一種推廣,是將變換中心移到視圓心以外的圓內的地方,馬上我們得到一個結論「反演半徑會隨著動點而改變」,接著,我們實驗了一下反演變換用有的一些性質,保角性,保圓性,…等在這個變換視中是否依然存在;接著我們用第二種方法來推廣反演變換,我們將邊界的形狀由圓視改成別的形狀(如三角形,四邊形…等等),然後也試試看在這種變換之下是否還擁視有反演變換的一些性質;第三種推廣,則是在研究的過程中,我發現了一種新的幾視變換,承接第一種推廣,我們將原先為定點的變換中心改為動點,將原先的動點改為定點,做出來的一種新變換。In the study, a new geometric Inversive transformation through three points is discovered. Here is the main result：（1）The first, onto cycle of inside and outside can be proved under invasive transformation. It is changed moving the center from center of cycle, we can get a new ” Inversive radius can be changed by moving drop. (2) We hope to find the answer to this problem by experiment, it is exist with the inversive properties. (3) A new geometric transformation is discovered, a fixed drop can be changed moving drop, then the first moving drop shifted the fixed drop. This leads to a new construction if the new transformation.
別鬧了，辛普森先生
We investigate the machinery producing successive Simpson’s paradoxical reverse. Taking advantage of algebraic and geometric techniques, we obtain the following results. Take playing baseball for example. In our study, we find that Simpson’s paradox only occurs when the hitter’s hits over 3 times in one game. Set n equal to the times I will hit in one game. If my batting average in each game is at least(n ?1)/2 times higher than the others’; then I am sure that my total batting average would not be invert by the others. In order to find how many the lattice points in the triangle, we use Pick’s formula. But sometimes, the Pick’s formula is not appropriate to triangles whose vertex are not all lattice points. So we develop New Pick’s formula to estimate the number of lattice points in such kind of triangles. Besides, we also find an iterative algorithm to produce successive “Simpson reverse” phenomenon by using C++ language, and we can therefore produce as many “Simpson’s set of four sequences” terms as we like(not beyond the computers’ upper limit).Moreover, if both sequences of ratios converge, then they must have the same limit.我們探討了一般人乍看之下顯得頗弔詭的辛普森詭論。我們配合GSP 作圖，用解析幾何、設立直角座標系和C++ 程式的運算，找出在特殊情況下或一般情況下所產生的辛普森數列組和特殊的性質，並且以棒球場上的打擊率為例子來做印證。通常一場棒球賽中，每個人平均上場3 次～4 次，經過我們的討論，發現要發生逆轉的機會只有在上場達到4 次或以上時才會發生。?了求出在直角座標系中可以滿足的格子點個數，我們用了Pick公式，但?了更準確的估計，我們引進了虛點的概念，重新推導出了新Pick 公式。另外，我們還發現，假設兩個人上場比賽，若打了2 場，且每場最多上場打擊K 次，其中的一個人的打擊率只要是另一個人的(k-1)/2倍以上就保證不會被逆轉。我們又找到了連續產生辛普森逆轉的演算法，利用C++ 寫出程式，經由演算法和遞迴式，製造出項數可任意多(只要電腦能夠承受)的辛普森數列組，且我們發現若兩個比值數列接收斂，則極限趨近於同一個數值。
摺紙數列-相關問題探討
1. 遊戲規則：將1～ 2m × 2n的連續正整數，由上而下、由左而右依序填入 2m × 2n的方格內。操作規則允許將2m × 2n做往右或往左或往上或往下的完全對摺，直到操作至所有單位方格均疊成一行，此同時有數字也由上而下形成一數列。2. 本研究即是探討操作完成的數列之數量與數字間的關連性。3. 我們發現：(1) 數列之數量與巴斯卡三角形有關。(2) 形成的數列必符合內文的 [ R(L) 性質]、 [ D(U) 性質]、[ R&D 性質]、[D&R 性質]。
1. Rules of thegame: Fill in order the continuous positive integers 1~ 2m × 2n, from top to bottom and from left to right in the 2m × 2n check. The operational rule allows a complete fold of 2m × 2n either rightward or leftward, or upward or downward, until all the check units pile up in a line. At the same time, all the integers form a series from top to bottom. 2. This study explores the relationship between the number of the series and the integers after the operation. 3. Our findings are: (1) The number of the series is related to Pascal triangles. (2) The series formed meet the properties mentioned in the study: [the property of R(L)], [the property of D(U)], [the property of R & D], and [the property of D & R].
由Brocard Point 發現幾何不等式
本研究報告以Brocard Point 為核心，所用到的性質均先證明，以確認其正確性，並推演出一些其他的性質，藉由這些性質導出幾何不等式。內容可概分為四部份：(1)以Brocard Angle 及已知的或推演出的基本性質，導出一些不等式。(2)結合「法格乃諾問題」、「費馬點」、「尤拉公式」導出幾個幾何不等式。尤其是三角形邊長與面積，外接、內切圓半徑與邊長間的不等關係，頗為有趣。(3)以向量為工具，分別計算內、重、垂心與Brocard Point 間的距離，並導出邊長的不等關係。其中由內心及重心所導出的不等式，清楚俐落；垂心所導出的不等式則較為複雜。(4)以Brocard Cirle 與內、重心間的關係，導出一系列的不等式。其中Weitgenberk 不等式的無意發現，令我們印象深刻。The Discovery of Geometry Inequalities by Brocard Point This paper takes Brocard Point as a core. We proved some properties about Brocard geometry to confirm its accuracy, and deduce some other properties, and then derive some geometry inequalities by these properties. The content may divide into four parts: a) Derives geometry inequality by Brocard Angle, Crux Mathematicorum and properties which known or deduced. b) Unifies "Fagnano problem", "Fermat Point", "Euler formula" to derive several geometry inequalities. In particular the inequalities between triangle area and length of side, or circumradius inradius and the length of side, is quite interesting. c) Derives geometry inequalities about length of sides in triangle by the distances between incenter centroid circumcenter and Brocard Point. Especially, these inequalities were elegant which derived by incenter and centroid, but it was complicated derived by orthocenter. d) According to the relation about incenter centroid and Brocard Circle derives a series of inequalities. Discover Weitgenberk inequality makes us excited.
Quantitative Analysis of Organism Growth Using Fractal Dimension Statistics
Cultures of bacteria were analyzed using fractal geometry and statistics to provide a method for predicting organism growth, paving the way for a better design of treatment drugs. Images of three cultures of isolated Bacillus subtilis were taken at time intervals of two to three hours for eight days. The images were processed using the IDOLON program and quantitatively described using three statistical formulas: fractal dimension D, Renyi dimension and Hausdorff-Besicovitch dimension. The three variables were integrated to compute the maximum of the distribution and were used as coordinates for a 3-dimensional graph f. A 2-dimensional graph g containing the maximum of a distribution under time analysis was also constructed. Topological properties of the graphs, including slope, direction and area were used to determine the interrelationship of the three fractal values. The two graphs, described as φ - : X -? P1 where X is the smooth algebraic assimilation of the four variables under time analysis, was extended using Java. A computer-aided prediction model of the graphs f and g were made which combined the topology of f and g at infinity. The computed fractal values showed the existence of a fractal pattern in the growth of Bacillus subtilis with fractal dimension ranging from 0.900 to 4.000, indicating a linear iteration. This was supported by the values of the Renyi dimension, which showed a horizontal growth pattern of the bacterial cultures, establishing the growth of the bacteria to be inclined to go towards the North East direction. There was consistency in the computed fractal values, maximum of distribution and topographical computations of all three cultures which also indicated the existence of a pattern of growth which could be extended to tinfinity, thereby allowing prediction of the direction and rate of growth of the bacterial colonies. The fractal patterns in the growth of bacteria, in this case Bacillus subtilis, yielded the direction and rate of growth of the bacteria as shown by the analysis of the fractal patterns and statistical values, showing that the growth of harmful organisms can therefore be predicted, making it possible to improve on the design of drugs for the control of perilous cells. By preventing the growth of insidious cells, the potential effects of virulent organisms may be avoided, and treatment may be made more possible.
再論巴斯卡三角形
本研究以b04課程中的巴斯卡三角形為研究對象，將原先巴斯卡以「1」為首、「+」為運算符號的規律三角形，改為以「-1」及「ω 」為首、「×」為運算符號，分別就其產生的新三角形作探討，發現其中似乎隱藏著原先三角形所沒有的規律性。為了更瞭解這種規律，藉由電腦軟體繪出其圖形，圖形顯示出如碎形般的複製關係，不論放大或縮小，其中的遞迴關係並未改變，頗令人好奇，因此著手研究。研究過程中對於圖形的規律性採用先臆測、接著歸納、最後給予證明的方式呈現。得到以下的結論：一、分別以數列呈現新三角形圖形的規律性。二、分別將新三角形中每一列中的某數字（如-1、ω 或ω 2 ）的個數予以通式表之。三、分別推算出新三角形第n 列第j 行的數是「1」或「?1」及「1」或「ω 」或「ω 2 」。四、相同的模式，在特定的圖形範圍中，不斷重複出現。許多研究將巴斯卡三角形中的所有數，以某數為模的餘數紀錄下，去探討其餘數在新產生的巴斯卡三角形中的分布情形；而在碎形的研究中，大部份著重如何畫出碎形。本研究著重圖形其規律性的探討，提供上述研究不同角度的詮釋與探討。 This research subject is based on Pascal’ s triangle in senior high school curriculum. The regular triangle begins with「1」and use「+」as operation. Let 「1」 be replaced with「-1」and「ω 」, the operation sign「+」be changed into「×」. I do research on the new triangle and discover the seemingly hidden regularity which doesn’t exist in the original one. To understand more about this regularity, I draw figures through the computer. The figures show the relationship of reproduction as fractal. Whether the figure is enlarged or minimized, it’s surprising curious the recursive relationship doesn’t change, so we begin to work on research. In the process of the research, we make careful observations, assumptions and deductions about the regularity of the figure. Finally, we come to some conclusions by means of giving proofs:（1）Present the regularity of the new triangle figure with progression.（2） Present such numbers as「-1」, 「ω 」, 「ω 2 」 in each row of the new triangle with formulas separately. （3）Figure out the number in the row n and in the column j of the new triangle is「1」or「-1」,and「1」or「ω 」or「ω 2 」. （4）The same model appears again and again in the specific range of figure. Many researches record Pascal’s triangle modulo certain number to explore the distribution of remainders in the new triangle. In the research of fractal, how to draw fractal is mostly focused on. The exploration of this research emphasizes the regularity of figure, offering the interpretation and exploration of researches above from different angles.
殊途同歸-格子點平面最短路徑和之探討
本研究從理想城鎮(Ideal City)街道開始，討論平面上相異n 點到某一點的最短距離和。經研究後發現：當n 為偶數時，則到相異n 點的最短距離和所形成的區域可能是一個點、一個線段或是一個矩形；當n 為奇數時，則相異n 點的最短距離和所形成的區域將會退化成一個點。此外，本研究將理想城鎮的街道換成正三角形的街道幾何平面，同樣是討論平面上相異n 點到某一點的最短距離和。經研究後發現：當n 為偶數時，則相異n 點的最短距離和所形成的區域可能為一個點、一個線段、一個四邊形、一個五邊形及一個六邊形；當n 為奇數時，相異n 點的最短距離和所形成的區域則可能為點、三角形的情況。假使考量各點重要性的比重，分別加權後再求最小點。研究發現無論在理想城鎮或正三角形幾何平面上，皆可將各點視為多個權數相同之點重疊於此點上，便可利用先前的方式求得最小點區域。透過這次的研究，可以利用n 個相異點到某一點的最短距離和實際應用在貨物運送的問題或是消防設施配置等問題。The present study was intended to start with the Ideal City and proceed to discuss the sum of the shortest distance between a point and n different points on a plane. After the discussion, it was found that if n is even, the formed region could be a point, a line segment, or a rectangle. If n is odd, then the formed region must be a mere point. Further, the current study transformed the Ideal City into the geometric plane of an equilateral triangle. Similar to the previous discussion, if n is even, the formed region could be a point, a line segment, a quadrangle, a pentagon, or a hexagon. On the other hand, if n is odd, then the formed region could be a point, or a triangle. The result of this study, which investigated the sum of the shortest distance of a certain point to n different points can be applied to the real life situation, such as transporting goods or distributing fire control facilities.
生生不息-正五邊形的繁衍及算術法則
This study was to explore the nature of two basic constitutes of the regular pentagon,With these two constitutes, the regular pentagon could be multiplied into any times in size. We used four multiplication methodsto show how the regular pentagon enlarge and to verify that the enlarged regular pentagons derived from computer did exist. By integrating these four multiplication rules, we were able to arrange regular pentagon of any length of side, and evidenced the equation was ( If m,n is the number of A,B of a regular pentagon respectively ) When we tried to verify if any regular pentagon could be constituted by other smaller regular pentagons, we found that it was un-dividable only if the length of pentagon side were (the number of A, B were the 2n and 2n-1 item of Lucas Sequence), otherwise, any regular pentagon is able to be constituted by other smaller regular pentagons. The divided forms could be multiple. We also found that any pentagon could be divided by two successive un-dividable pentagons, which is called “standard division rule”. We expected to derive all kinds of division by analysis of two successive un-dividable pentagons in standard division rule.
這個研究起源於一個拼圖玩具：利用兩種黃金三角形排出指定大小的正五邊形。我們的研究動機是：一、 假如無限量供應A 和B，能夠拼出哪些邊長的正五邊形？二、 哪些拼好的正五邊形不能拆成一些較小的正五邊形？我們將研究的主要結果分述如下：