在高二學數學的時候，老師曾提到共軸圓系有所謂極限點的名稱 ,但當時對於極限點的求法並未詳細介紹，我們為了進一步了解，便在蔡安華、郭茂雄兩位老師的指導下，根據共軸圓系極限點的定義，導出一般共軸圓系極限點的求法公式，同時也探討有關極限點的一些性質，更將這種方法推廣以達三度空間共根面球系時，發現也有相同關係，雖然這項結果也許不是新的，但是在探討的過程中，卻也提供了我們學習方法的一項模式。

Our research started from June, 2003 to May, 2004. During these twelve months, we collected and discussed the basic ecology material of malformed “ Rana latouchii ”, which we call it frog in the following paragraphs . Thanks for the favorable geographical position, we collected the first-hand data in the area of Jin-Mian Mountain foothill gully situated in Taipei. Our investtigation project includes：the percentage of the malformed frogs in a whole race, the geographical distribution of dominant-abnormal frogs, the sexual distribution of dominant-abnormal forgs, and the possible roots for the generation of malformed frogs. We have altogether carried seventy-six times field investigations. In those investigations, we focused our research on the randomly-sampled 580 Rana latouchii. In these 580 sample, there are 190 female, 350 male, and 40 adolescent. We discovered seventy-six malformed frogs, constituteed by 21 female, 53 male and 2 adolescent. In another words, the occurance of malformed frog in a race is around 13.1. %. In our observation, we can classify dominant-abnormal frogs, “Rana latouchii”, into eight species. Here are the species: 1. the entire body is obviously malformed, 2. with only one eye, 3. in lack of appendage apod, 4. in lack of arms,5. in lack of palms, 6. with excess webbed toes, 7. the maltfromed of appendiculars, 8. the lack of toes on the palms. Within these 8 species, the last species has the highest occurance, 66.38% in aproximation. If we look at the classfication by sex, the ratio of female to male is around 1 to 2.52(1:2.52). However, the possibility of being maleformed is regarded as the same for female and male frogs in the same group. In statistical term, our research shows that the chi-square is 1.742 while our degree of freedom is 1 and confidence level is 95% (X2=1.742，df=1，p>0.05). During the research, we also observe that each malformed species affect different living activities of malformed frogs. In the worst situation, the malformed nature can result in shorter life. In our sampling area, malformed frogs mostly reside in high concealment and high humidity location. In our research data, within these area, our re-captured rate for malformed frogs is 25% and hirudin-parastical rate is 4.31%. As our research shows, the occurance of malformed “Rana latouchii” in our selected area results from 3 main causes： First, the parasitism of leeches or parasitic worm；Second, the agriculturally chemical contaminants；Third , the injured causes from escape from predator. 本研究自2003年6 月至2004年5月為止，共12個月的期間，於臺北市內湖區金面山南麓溪谷，對拉都希氏蛙(Rana latouchii)族群之畸形蛙個體進行基本生態資料的蒐集與探討，調查項目包括：族群中畸形蛙發生的比例、畸形蛙外部形質異常分佈的情形、族群中雌蛙與雄蛙發生外部形質異常之情況及可能造成畸形蛙產生的原因。 研究期間共進行了76次的野外調查，總紀錄了拉都希氏蛙580隻，雌蛙有190隻，雄蛙有350隻，幼蛙有40隻；畸形蛙個體共紀錄有76隻，雌性畸形蛙有21隻，雄性畸形蛙有53隻，幼體畸形蛙有2隻。族群中畸形蛙發生率約為：13.10％，外部形質異常的畸形拉都希氏蛙總計有八大類，包括：1.整隻個體形質畸形，2.單眼個體，3.附肢缺少，4.附肢缺臂，5.附肢缺掌，6.腳掌多趾，7.附肢腳掌畸形，8.附肢腳掌缺趾，其中以附肢腳掌缺趾類的66.38％發生率最高。畸形蛙中雌雄比約為1：2.52，族群中雌蛙與雄蛙發生畸形的比例並無顯著的差異（X2=1.742，df=1，p>0.05）。 不同部位形質畸形的發生，將會影響畸形蛙的個體活動，甚至可能降低畸形蛙的存活機率。研究調查樣區中，畸形蛙較常活動於濕度高且隱蔽度大的分樣區中，畸形蛙的重複捕捉率為25％，被水蛭寄生的比例則約是4.31％。 由調查研究結果顯示，造成內湖金面山區南麓溪谷拉都希氏蛙畸形蛙發生的主要原因，推測可能有下列幾種：(1)水蛭或是吸蟲類的寄生；(2)農作過程中的化學污染物；(3)逃脫掠食者捕食過程受傷。

這個專題研究的目的是要發展出一套簡單可靠的方法和廉價自製的器材，在地面上即能有效估測大氣層的厚度。我們小組研究光學中雷氏(Rayleigh)散射的原理，針對空氣分子對光線散射作用和特定方向之偏極效應，利用一已知散射長度之路徑，測量其偏極光的強度，同時比對由大氣層散射而來，在同一偏極面上的散射光強度，即能估算大氣層的厚度，方法簡單新穎，自製器材經實際測量和改進，有發展和推廣的價值。\r The main idea of the experiment is to set a system in order to effectively estimate the thickness of the atmosphere. On the theory of “Rayleigh Scattering” (small air molecules sizing about 10-4μm), we developed an equipment that has two tubes. The tubes lead the scattered lights from two paths. One is called “air light” scattered in the ground air, and the other “sky light” is scattered in the sky and reflected by a beam splitter. The two paths are on the same plane; the scattered lights are perpendicular to the direction of sunlight and 100% polarized. We could adjust and measure the distance “d” of the air light path. We simultaneously observe and compare the intensity of the lights from the two paths with the electronic instrument made by ourselves. By using the known distance “d” and the reflection “x” of the beam splitter, we can calculate the thickness of the atmosphere. The experiment is simple, novel and easy to do in an extensive field at school. Researchers don’t have to use a bloom, radar or satellite to discover the atmosphere, but you could use a simple equipment to observe the features of it.

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.

在去年寒假中我們曾經到過台北桃園一帶的魚池去垂釣，發現到魚體上常有許多白色毛狀物。請教老師以後才得知該魚是患有水黴病。因該病發生的時間多數在冬末春初至四、五月間，也正當魚卵孵化稚魚大量繫殖的季節，所以常見有魚卵及稚魚因水徽病的發生而大量死亡，對國民經濟影響甚鉅，為了要更瞭解水黴病對魚體影響，因此我們便開始作這方面之研究。

本研究發現：\r 一、 液滴滴入水中產生甜甜圈結構的過程有六個時期：\r 1水面反彈。 2杯狀期。 3甜甜圈小環期。\r 4甜甜圈大環期。 5滯留期。 6無序散逸期。\r 二、 液滴滴入水中產生甜甜圈結構的影響因素有：液滴落下時的高度、液滴的濃\r 度、溶液的濃度、水溫、水面擾動、液滴大小、液滴之初速……\r 在本研究的觀察中，有幾點結論：\r (一)當高度差=0cm 時，都可以形成甜甜圈。\r (二)滴液濃度為8%及10%滴入純水中，比較容易形成甜甜圈。\r (三)密度小的滴入密度大的溶液或密度相差太大時不適合甜甜圈結構形成。\r (四)在不同容器中觀察產生甜甜圈結構過程，對實驗結果並無顯著之影響。\r (五) 改變水溫會使觀察的情況更複雜。

本實驗以薄膜色層層析（TLC）、高效能液相層析（HPLC）分析等化學方法，進行金銀花品種差異的鑑識；此外，配合生藥學的顯微鏡檢視，如中藥材組織鏡檢、藥材粉末鏡檢等比對，以期找出辨別金銀花品種的方法。研究至目前為止，由金銀花之薄膜色層分析的Rf值（0.225、0.425、0.7、0.85、0.95）可確認出金銀花藥材，並得知金銀花藥材中皆含有綠原酸的成分；以高效能液相層析的圖譜與成分峰的積分面積可用來判別金銀花的品種，並從質譜分析瞭解成分含量；進行生藥學的藥材粉末組織鏡檢，發現無法作為金銀花藥材的分類憑藉。未來將持續延伸實驗，朝向中藥奈米化與一般粉末在藥效、成分上差異的比較，並進行金銀花萃取液的抗菌作用試驗，瞭解不同品種之金銀花藥材在藥理作用的異同，接續著奈米化藥材的應用與實踐。Using thin layer chromatograph (TLC) and high performance liquid chromatography (HPLC), we can study how to differentiate the species variation of honeysuckle; beside, based on the observation of biopharmaceutical microscope, such as comparing the histology of Chinese herbs and its powder, we suggest that we could differentiate the species of honeysuckle. From the present, firstly, we could distinguish the honeysuckle from other herbs by the Rf value of TLC(0.225, 0.425, 0.7, 0.85, 0.95), from which we find that all honeysuckles contain the component of Chlorogenic acid. Secondly, we could tell the species of honeysuckle according to the map of HPLC and the peak area after integration, as well as the integrants of honeysuckle by way of LC-Mass analysis. Thirdly, while studying the histological analysis based on the observation of biopharmaceutical microscope, we found that it shows no difference between all the honeysuckles; thus, it fails to be a scientific method used to distinguish the herb honeysuckle. However, in the biochemical experiments of honeysuckles, we found honeysuckles from different sources and the place of origin shows difference in their antibiotic effect, showing the importance of local medicine. When it comes to my future work, in order to extend my experiments on honeysuckles, I would compare the nano-scale honeysuckle powder with normal-sized one in their clinical effects and components.

偶然的機會，在木柵動物園蝴蝶館步行道側的人造小溪內發現許多會移動的小葉片，詢問之下，原來裡面躲藏的竟是一隻可愛的小生物。牠亦是石蠶蛾的一種，由於好奇，我們決定探討牠的築巢行為，在長時間的觀察中，發現亦有相關的本能行為出現，故設計下列實驗作一系列的探討。

由於蜘蛛絲複雜的分子結構及產生過程，長久以來一直被視為一個特殊的生物材料(高延展性，高韌性，和高強度)，此外，前人研究指出蜘蛛能自己調控絲的性質，並受到環境的影響。然而，大多數的研究多以結網性蜘蛛為主，只有極少數研究著重在探討非結網性蜘蛛，如:跳蛛。本研究中，以安德遜蠅虎為材料，分析跳蛛的跳躍行為，以及探討曳絲在跳躍過程時所造成的影響。我們初步的研究結果顯示: (一)曳絲在跳蛛跳躍過程中，對於安全降落扮演重要的角色，及(二)跳蛛會藉由改變身體的角度來維持身體的平衡。在跳躍過程中，蜘蛛的跳躍速度會因空氣阻力而減少，但是蜘蛛絲的彈性恢恢復力(根據虎克定律)會讓跳蛛跳躍速度更顯著的減少，並藉著身體的轉動與曳絲的作用達到身體平衡。對於一個非結網性蜘蛛是另外一個不可或缺的輔助工具。相對於其他跳躍動物，有絲的跳蛛具備另一項能減緩降落速度的工具以增加降落的安全性。

這個研究選擇學校及住家附近花圃，常見的蝸牛為對象，來研究蝸牛回到原點的行為與機制。研究1發現並非所有的蝸牛都會回到原點，而回到原點的機制與食物及環境有關。研究2發現蝸牛是在找尋食物，扁蝸牛會棲息在食物附近而高音符絲鼈甲蝸牛也要求潮濕陰暗的環境。研究3發現蝸牛有逆趨地性、逆趨光性、趨濕性，這都會影響蝸牛往哪裡去。研究4發現蝸牛沒有聽覺，視覺不佳，沒有分辨顏色的能力，靠嗅覺來感知及分辨外界的訊息。研究5發現蝸牛具有痕跡跟隨的現象，而痕跡能遺留很久，讓蝸牛可以依循而找到原來的環境或同伴。研究6發現高音符絲鱉甲蝸牛沒有明確的記憶能力。

本研究在探討長腳捷蟻的生態及防治方法，研究地點在本校園內，研究時間從2017年8月底到2018年4月，共進行9個月。長腳捷蟻是屬於遷徙性高的螞蟻，蟻巢中有多蟻后，特徵是明顯的長腳與觸角以及在受到干擾時的瘋狂舉動（也因此有狂蟻的稱號）。驚擾時的爬速每秒移動約5.40cm;其蟻橋間隙的極限是17mm，偏好吃食有受傷的食餌，較快被搬走。長捷的攻擊模式會先用大顎咬，再舉起腹部噴蟻酸攻擊，會多隻輪流噴蟻酸攻擊，被噴蟻酸的螞蟻最後會死亡。同種間不同區域的長捷都無互相攻擊;也無爭食現象;也會進入不同區域的蟻巢口，我們推論本校區長腳捷蟻的族群是一個龐大的超級聚落（Supercolony）。以3%的硼砂糖水濃度防治效果最好，驅離效果以10%的明星花露水水溶液最佳。

師教我們玩了一個圍棋的數學遊戲-「兵臨城下」，看似公平的遊戲，其實暗藏玄機。利用倒推法，我們一盤一盤的進行分析，終於在努力研究和老師的討論中，找到了其中的奧祕：（一）、一路玩法必勝絕招：掌握n＋1這個必勝點，就要掌握到n＋1的倍數，即可掌握優勢取最後一顆而致勝。（二）、二路玩法必勝絕招：掌握到對稱的盤型，或利用餘數判斷法，使兩路餘數相等。（三）、三路玩法必勝絕招：利用餘數判斷法掌握五個關鍵必勝點：（0,0,0）（0,1,1）（0,2,2）（0,3,3）（1,2,3）。（四）結合數學二進位法的運用。（五）進行遊戲的改良和設計：除可增加為n路玩法外，雙方還可以任意先擺好棋子，再開始互相進攻，以增加遊戲的趣味性和變化。