四面體體積平分面的包絡方程探討
剛開始考慮平分物件時，我們從二維的多邊形部分著手，後來發現已經有人做過相關研究，並且得到類似的結論。這個部份顯現出面積平分線與其包絡曲線間的密切關係。我們將其中的方法和結果加以歸納、改善，為了更全面地研究，我們推導出一般性的包絡方程。之後當我們推廣到三維領域時，發現四面體體積平分面與之前的結論有些相似之處，平分的情況卻也更複雜，我們將推導的結果用電腦軟體呈現出來，以便更深入地了解它。最後嘗試了相當抽象的高維積平分，結果仍具有工整的對稱性，讓我們充分領略了數學之美！When considering bisecting a subject, at first we focused our attention on 2-D case, polygons. But afterwards, we found there were already some similar studies conducted by other students, which indicated the close relation between the area-bisecting lines of a polygon and their envelope. We rearranged their methods and results, and then made further improvement. Moreover, in order to study the bisecting problem entirely, we derived the general envelope equation. Then when extending the generalization to the 3-D case, we came to the conclusion that tetrahedrons’ volume-bisecting planes is similar to that in 2-D, but the circumstances are more complex. We tried to show our result with the aid of software, hoping to understand it fully. Finally, we tried to do the case in higher dimension, which is very abstract, and the result was clear-cut symmetrical. During the studying process, we had seen “the beauty of mathematics.”
黑暗的力量
We study the magic power of dark energy and dark matter by using theoretical derivation and numerical simulations. We found that: 1. The dark energy will gain kinetic energy from the moving dark matter through gravitational interaction. Due to the law of energy conservation, the motion of the dark matter will slow down and satisfy Ek(t)−Ek0 ∝ρDE1.92 t, where Ek(t) is the kinetic energy of the dark matter, Ek0 is its initial kinetic energy, ρDE is the energy density of the dark energy, and t is the time. 2. The formation history and the structure of galaxies will be different due to the existence of dark energy. The more the dark energy, the earlier the formation of the galaxy core. In addition, the kinetic energy Ek(R) as a function of R will be different if the ρDE is different. Thus we can observationally measure the Ek(R) of galaxies, compare it will our results here, and then deduce the ρDE in our universe. The results here can be applied to the observations in the near future.
我們藉由理論的推導，配合電腦模擬的手段，來探討宇宙中黑暗物質和黑暗能量的神祕力量。我們發現：一、黑暗能量會透過重力交互作用而從運動中的黑暗物質獲得力學能，而且因力學能守恆，致使黑暗物質的速率減慢，滿足 Ek(t)−Ek0 ∝ρDE1.92 t, 其中 Ek(t) 為黑暗物質的動能， Ek0 為其初始動能， ρDE 為黑暗能量的密度，t 為時間。二、星系的形成過程及結構，會因黑暗能量的存在而改變。黑暗能量越多時，星系的核心會越早形成。而且動能 Ek(R) 隨著至星系中心距離 R 的變化，會因 ρDE 的不同而不同，因此可以試圖量測宇宙中星系的 Ek(R) ，然後和這裡的結果比對，即可推導出宇宙中的 ρDE 。這些研究成果，將可直接應用在未來的觀測結果上。
A Physical Analysis of the Difference in Ungues of Insects on Types of Habitat
1. Purpose of the research While studying about insects for school club activity, we found that there are differences in several anatomical characters of the insects according their habitats. Especially, the unguis was different as whether the insect lives in water or on land. So we observed the structure of unguis of some insects by microscope, and physically analyze to relate with the habitat of each insect. 2. Procedures First, we read papers and books about insect morphology to study about insect's unguis. Then, we collected samples of Chironomus plumosus(larva), Neuronia regina(larva), and Carbula humerigera. After pretreatment of samples, we put them in the SEM (Scanning Electron Microscope), observed the unguis of each insect and took pictures. 3. Data First, the larva of Chironomus plumosus has prolegs with numberless hooks that has a certain arrangement and the same angle of 90º. They also has several tiny swellings around the hook. Next, the larva of Neuronia regina has pincer-like unguis which were sharp and bend, reminding the shape of a quadratic curve. Lastly, Carbula humerigera has two large, thick pincer unguis, its form same as the of Neuronia regina. The unguis are very sharp and faced towards the land. 4. Conclusions Both the aquatic insects and the terrestrial insects have structures in unguis developed to increase precision on land. Especially, the angle of unguis were all close to 90º. In addition, aquatic insects like the larvas of Chironomus plumosus or Neuronia regina have particular characters increasing friction force according to resist the flow of water.
模擬複雜系統的演化
複雜系統廣泛地存在每個人生活的周遭，儘管這些系統在表象上有明顯的差異，卻不約而同的都呈現出臨界現象。一個簡單的物理概念卻能廣泛地使用在許多不同的場合當中，這是令人始料未及的。這份報告在尋找，是什麼原因導致這許多似乎與物理沒什麼關聯的系統，都呈現出臨界現象？以及這類系統是如何演化的？臨界現象在系統的演化歷史中，又是扮演怎樣的一個角色？一開始，先介紹臨界現象在實際系統中表現的形式，在這以文字系統作為範例。接下來，進入思考的階段，猜測系統個體間存有的關聯性，提出如何影響的假設，並建立模型。從模型結果，可以了解複雜系統隨時間演化的趨勢改變，並發現臨界現象在系統中扮演的角色以及代表的意義。這份報告有三項結論：1. 臨界現象在複雜系統中呈現的形式 2. 模型結果與實際系統十分相似 3. 臨界現象代表著系統的穩定態。The theory of Scaling Law and Universality was originally used in researching Critical phenomena, and now we find it existing in our normal life. A simple physical concept can be used extensively in different courses. This study try to find out the relation between individuals which causes Critical phenomena appear in such different courses in our normal life, and figure out the effect time brings about. To begin with, we introduce the sights of Scaling Law and Universality by exploring the formation of complicated system. Facing a complicated word system as a real example, we observe an extraordinary phenomena, and find out the Universality and Scaling Law in different kinds of linguistics. In the main part, to think further, what’s the relation between the individuals that leads to Critical phenomena? We will give hypotheses and build a model to simulate complicated system from the view of Universality, trying to find out the cause of the interaction of complicated system. From this model, we could understand: When a complicated system evolves with time, the Critical phenomena will naturally occur in this system that could be considered as dynamic equilibrium. From this study we found out(1) The Universality and Scaling Law in complicated systems, here we use linguistics as an example.(2) Under the hypotheses, the simulation is almost as same as the real result of linguistics system.(3) Time plays an important part in the Critical phenomena which can be found in many different complicated systems, and Critical phenomena symbolize a stable state of such systems.
明察秋毫-動態測微器
The purpose of this research is to create a device that is able to precisely measure small dynamic changes which cannot be recognized by the human eyes. The Vernier Caliper and the screw micrometer are common tools used to precisely measure lengths of objects. However, things which are measured by the Vernier Caliper or the screw micrometer have to be in a solid state, and the shape cannot be changed. By applying the light lever principle on Lego bricks, this research uses the LabVIEW graphical programming system to design a device which is able to automatically measure small dynamic changes. The precision of this device is higher than that of the Vernier Caliper and the screw micrometer. Moreover, this device is able to precisely detect the small dynamic changes of solids and liquids as well. Through numerous tests, the least count of the device can reach the level of 10-3cm. Also, this device has successfully measured small changes, such as the height of the liquid surface by one drop of water, the evaporation of water in one minute, and the growth of a plant in one hour. By popularizing this device, people will be able to precisely measure small dynamic changes which are difficult to be measured in a short time.