費瑪也瘋狂-平面上存在障礙時連接三定點的最佳網絡問題
在一個有障礙的平面上，給三個定點，我們探討連接此三點的最佳網絡。我們討論了諸如直線、射線、線段、圓、網格狀、三角形……等類的障礙，當網絡每穿越障礙一次，就必須付出代價，例如「拖延5 分鐘」。所以，設網絡穿越障礙的次數為y ，則網絡除了原本的總長度之外，還額外加入y 倍某固定數值的損耗。我們以費瑪點的各種性質及三角形不等式等方法為工具，就不同的穿越障礙次數綜合比較，而找出最佳網絡。在某些情況下，最佳網絡不是以費瑪點來連接三點，而是在障礙(如：直線)上找出符合某種與餘弦值相關特殊性質的點，以該點來連接三點，而此網絡可用GSP 軟體相當精確地作出。另外，我們也探討在考慮障礙造成損耗的情況下，兩點間的「實際距離」為何。 最後，我們考慮「混合障礙」問題。在此類問題中，除了前面所討論的障礙，還另加了如同「河流」的兩平行直線間區域之障礙，在這種障礙區域中，網絡的長度要乘以數倍來計算。我們發現，此類問題的最佳網絡也可用特定的正弦條件配合GSP 而相當精確地作出來。;Considering various kinds of obstacles in a plane, such as a line, a segment, a ray, a circle, a triangle or chessboard grids, which function like a red light, we research into the problem of finding the optimal network connecting three given points A, B, C in the plane amidst obstacles described above. Each time when the network crosses an obstacle, it will cause losses, such as five minute’s delay or a loss of one hundred dollars. Taking advantage of Fermat points, some basic inequalities concerning triangles and some special qualities about sine or cosine functions, we obtain the optimal networks in different situations. Besides, we consider what the “real distance” between two points is when there are obstacles in a plane. We also put another obstacle, including a line and a weighted region between two parallel lines, into consideration. In the region, like a river or a muddy ground in real life, the length of the network should be multiplied by a fixed time. Furthermore, we can use GSP to make the networks very accurately.
正N 邊形光圈之路徑追蹤
本研究是[對於正n 邊形A1A2…An邊上一點P(含頂點)，想像自定點P 朝鄰邊發出一條光線，若依逆(順)時針方向依序與每邊皆碰撞一次，經一圈而可回到P 點，則此路徑稱為「光圈」。過程試著追蹤在正n 邊形內能形成光圈的光線行進路徑及其相關問題。
本研究令，且以逆時針得光圈來討論：
1.根據[光的反射原理]，探討光圈之存在性，發現除定點P 在正2m 邊形或正三角形的頂點外，其餘皆有光圈。
2.將可形成光圈的路徑圖展開成[直線路徑圖]來探討。
3.由[直線路徑圖]，觀察到形成光圈的光線行進路徑，可能存在下列情況： (1)不通過正n 邊形的頂點，且產生路徑循環與不循環問題。 (2)通過正n 邊形的頂點。
4.發現正2m 邊形光圈皆為[完美光圈]。
5.發現正2m+1 邊形光圈之路徑與有理數、無理數之特質有關。即當s 值為有理數時，路徑會循環；當s 值為無理數時，路徑不循環。
The research is about [on Point P (including the angles) on the side of regular polygons A1、A2…An , imagine the light goes from Point P to the closest side, then bumps each side sequentially counterclockwise. After going a circle, it’s back to Point P. The track is called “the circle of light.” I try to trace the light track of the circle of light and other correlative questions.]
In this research, we suppose,and we discuss the circle of light according counterclockwise direction:1.According to the light reflective principles, we discuss whether the circle of light exists or not. And then we discover that the circle of light really exists except when Point P is on the angles of regular triangle or regular 2m polygons. 2.Spread out the circle of light’s track to [rectilinear track.] 3.By [the picture of rectilinear track], observing there are two kinds of the circle of light’s track: (1)If the light doesn’t go through the angles of regular polygons, it can be a circulative track or a non-circulative track. (2)When the light goes through the angles, it stops. 4.We discover that all the circles of light in regular 2m polygons are [the perfect circles of light.] 5.We discover the circle of light’s track is correlative with rational numbers and irrantional numbers. When s is a rational number, the track is circulative, if s is a irrantional number, the track is not circulative.
M&m Sequences 之研究
本專題的目的是研究以任意實數 a1 、 a2 、 a3 為起始的M&m Sequences 之穩定性質。我們主要關心的問題是：(1) 是否任給定三數a1 、 a2 、 a3 為起始的M&m 數列皆會穩定？(2) 若上述的M&m 數列穩定，則其穩定的長度與a1 、 a2 、 a3的關係為何？(3) 其穩定的值與a1 、 a2 、 a3的關係為何？我們研究的主要步驟及結果如下︰1. 當1 2 3 a 1) 為起始的M&m 數列。3. 我們證明了下列性質：(1) 若M&m 數列中前n 項所成數列的中位數為n m ，則下式成立： (2) 當存在 k > 4 , k ? N ，使得 ?1 ?2 = k k m m 成立時，則此數列穩定，且穩定長度p 滿足：min{ | 4 } ?1 ?2 = > = k k p k k 且m m ，其中p 必為奇數。(3) { n m }為單調遞增且, 5 1 ? ? ? a m n n n4. 如果x ? 41.625，則{?x,1, x}為起始的M&m 數列，其對應的數列有相同的大小次序且此M&m 數列會穩定，穩定值為41.625，且穩定長度為73。5. 我們觀察發現：如果x 1). 3. We prove the following properties: (1) If the median of the former n numbers of the M&m sequence is n m , we obtain (2) There exist k > 4 , k ? N such that ?1 ?2 = k k m m , then the sequence is stable and the stable length min{ | 4 }?1 ?2 = > = k k p k k and m m , where p must be an odd number. (3) { n m } is monotone increasing and , 5 1 ? ? ? a m n n n . 4. Suppose x ? 41.625, then the all M&m Sequences beginning with –x , 1 , x are the same, and the sequences will be stable, the stable value is 41.625 and the stable length is 73. 5. By the computer experiments, we observe that if x is any positive real number less than 41.625, the M&m Sequence starting with –x, 1, x, will be also stable but does not appear to follow any clearly discernible pattern of behavior. However, the stable lengths are much variant and exist some unknown relation with point format of x. Moreover, we have the following properties: (1)If x is a node, then the stable value is x and the stable length equals to the index of median of the node + 2； (2)Near the branch of 41.625, the stable length is almost a constant except at the edge area，the stable length of (-x,1,x) as x around branch 1 is chaos； (3)If x near the node (K= 3, 5, 7, …, 67, 69), then the stable length is l(K)+K?1 where the positive integral l(K) is determined by Prop1 (see Table 6 and 7).
鄒之風聲-風笛
「風笛」是台灣原住民鄒族的信號用具及祈雨法器，由一條繩子綁一支竹片構成。轉動風笛時，竹片會繞繩子自轉並拍打空氣而發出聲音，並有上下飛舞的現象。風笛產生聲音的原因，為竹片拍打空氣而造成的渦流共振現象；又由於繩子扭力大小及方向改變，使風笛的音調忽高忽低、響度忽大忽小、且竹片會在兩個平面上公轉，而有週期性變化。施力使風笛公轉轉速加快時，竹片自轉速率也變快，使其音調愈高、響度愈大；而繩愈短、愈粗時，竹片的公轉週期將愈短。The wind whistler was once used by Tsou aborigines as a tool for message transfer. It is composed of a string and a bamboo flapper. When swung around, the flapper spins, beats the air, and makes sounds. Moreover, the flapper flies up and down during the revolution. The spinning flapper beats the air, causes the vortex resonance phenomenon, and thus produces sound. As the twist torque and direction change, there is periodical variation in the sound volume, sound pitch, and the movement of the flapper, which orbits up and down at two planes. If given force to speed up its revolution, the flapper，s spinning frequency also increases, which makes the sound pitch higher and the sound volume greater. Besides, when the string is shorter or thicker, the flapper，s revolution period will be shorter.
新型空氣清淨燈具之研究與開發
本研究主要的目的是在開發同時具有空氣清淨與照明的兩種燈具。其中桌燈是基於自然對流原理，利用燈泡發熱讓氣流通過燈具上方的濾網達到過濾功能，為了尋求過濾效果與照度兼顧的最佳值，本研究並提出比較因子的概念。在吊燈方面，除了運用自然對流原理之外，還更進一步利用太陽能驅動風扇，進行強制對流，強化過濾的效果，使得本研究成果更趨於完善。 由實驗結果可得知，桌燈在四星期長期測試條件之下，其過濾效果增進率分別為39.1, 40.8與 40.1%。在吊燈四週長期實驗的結果方面，螺旋與 100W 鎢絲燈泡在自然對流的過濾效果增進率分別為49.1%與 51.4%，而100W鎢絲燈強制對流方面過濾效果增進率則為60.2%。由整個研究結果可以發現，本燈具對於空氣清淨有極佳的效果，在不增加額外耗能條件之下，能增加燈具的散熱效果與延長壽命，同時又具備空氣清淨效果，對環境空氣品質具有相當的貢獻。 The purpose of this study is to develop a novel lamp with both the functions of air-cleaning and lighting. One of it is the desk light. Basing on free convection principle, it makes the air run through the filter on the top of the lamp by its heat in order to attain the aim of air cleaning. To find the optimum value of both cleaning effect and illumination, we advanced the compare factor. The other is the droplight, though it is based on the same principle, we use the solar energy as its power to drive the fan. So that the effect of the filter can be augmented and the result of this research approach perfect. According to the experimental result, in the four-week experiment with desk light, the enhanced efficiency of filter is 39.1%, 40.8% and 40.1% respectively. On the way of droplight with four-week experiment, the enhanced efficiency of filter is 49.1% and 51.4% with helix and tungsten(100W) lamp under the condition of free convection; the enhanced efficiency of filter is 60.2% with tungsten(100W) lamp under forced convection. All these results of the research shows that the novel lamp has great performance on air cleaning and much better effect of heat sink without extra consuming of energy, also the lifespan of the lamp can be extended. Furthermore, it is capable of air cleaning and contributes to the quality of environmental air.
Fast Fabulous Flush
Water is very vital in our lives as we cannot live without it. However our world is now facing a serious problem. Owing to the increasing population, water resources are scarce. In recent years, we can see that droughts have been affecting millions of people around the globe. In the meantime, people in developed countries have been wasting huge amount of water for flushing the toilet. According to the US Environmental Protection Agency, 30% of household water goes toward flushing the toilet. Countries like China, US, Canada and UK are still using fresh water in flushing, consuming 190 million L of fresh water every day, not to mention the energy needed in pumping the water. In fact, it is not necessary to use so much water to flush away substances like tissue, hair, urine etc. The water we used is far more than we need. However, as we cannot control how much water is used when we flush, all water in the cistern is flushed away. Realizing the seriousness of water shortage and wastage in flushing, we tried to invent a device to conserve water by controlling the amount of flushing water used. Firstly, we study the principle of normal flushing system so as to understand why flushing cannot be controlled. Then, we tried to think of ways to control flushing. We have tried various methods and materials. After the 6-month testing and modification, we successfully invented Fast Fabulous Flush. It is a device which can be fit into existing cistern to conserve water. With our invention, users can control the amount of water flushed according to needs, so as not to waste unnecessary water. Our invention costs a low price which is no more than 2 US dollar. Also, it can be fit into existing cistern within 3 minutes with simple installation process. Most importantly, flushing water can be conserved effectively. It is estimated that around 200L of water can be saved per household every day.
水分子自我組裝之機制探討
Up to this time we have spent almost three years in studying condensation and water droplets. Little could we have done as compared with the almighty nature. However we are rewarded by the nature as we gradually found the secrets about electro-magneto field and water droplets: The size of water droplets turn smaller upon electro-magneto field and grow more uniformly especially upon electric field. This experiment presented here is actually the diary of the growth of water droplets in condensation, upon magnetic field and electric field. Through convection, it discusses the self assembly patterns of water droplets and peep into the uniformity both of the size and the distribution mode of water droplets. In former basic experiment, we focus on temperature and the speed of water moisture; generally speaking, higher temperature speeds up the coalescence procedure but does not affects the nucleation size of water droplets in simple plain surroundings; while speed of moisture does affects the nucleation size. As we went farther, deep into convection and found magneto-electric force did play an important role in the self assembly mechanism of water droplets. The topic is mostly concerned as we are surrounded by magneto-electric waves in today’s world. This experiment anchors the first step in discovering the uniformity of water droplets in different environment, and providing insights into the self assembly mechanism of water droplets upon electro-magneto field with nano sizes. 這是一系列關於水蒸氣冷凝為極細微小水珠的實驗。其中可以分為兩大部分； 第一部分是基礎實驗。將水蒸氣導入至潔淨的介面上(蓋玻片)，觀察冷凝水珠的結構。雖然看似簡單平常，但卻有令人驚奇的發現；不同溫度的水蒸氣，其冷凝最初始的細微顆粒之尺寸是相同的 ！爾後隨著溫度的升高，堆疊速率也跟著上升；以致於最後一起呈現出來的水珠大小不一，尺寸不一。 第二部分是將水蒸氣導到磁場及靜電場上，觀察其冷凝結構。這部分的實驗推翻了一般「水分子是電中性在電磁場中不受影響？」的刻板觀念 ！實驗所呈現出來的冷凝水珠，不但於附加磁場中尺寸縮小又不易長大，同時還有固定的自我組成模式( Slef-assembly pattern);而且也發現在磁場中的冷凝小水珠的尺寸比電場中的小，可是電場中的小水珠則表現出較大的均勻特質。