cuboctahedron

正八面体用纸怎么折

cuboctahedron 。

[英]kʌbɒktə'hedrən 。

[美]kʌbɒktə'hedrən 。

。

立方八面体

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配位数与配位多面体

我们在这里所要讨论的是由8个等边三角形组成的正八面体，每个顶点都有4个三角形相交于此(图1)，且其他的顶点也是如此．将图2放大，制作一个正八面体．边长8cm的三角形做出的模型大小适中，而且用一张A4的纸或卡片纸刚好．如果你是使用卡片纸，记得要在每条线上刻出印痕，才能折出整齐的边．。

我们可以从许多角度来观察正八面体，每一种角度都能使我们更了解它．从展开图建构模型，使我们的注意力集中在面的形状与在一个顶点相会之面的数目．但是当你做好模型后，正八面体的其他性质就显而易见了．想象一下将正八面体水平切成两半，切面通过A、B、C、D4个顶点，如图3，将正八面体切成两个相等而且以正方形为底的金字塔．如果将正八面体旋转，使得任何其他的顶点如A或B位于上方，则所得出的结果也会相同．事实上，如果正八面体上没有任何标记，要区分一个顶点与其他顶点的不同之处是不可能的；面的情况也是如此．。

由于这种对称性，任何通过一对相对顶点的二分切割都会得到如图4所示的正方形切面．。

这给了我们一种新的角度来观察正八面体，也提供了制作模型的不同方法．。

用卡片纸剪出两个正方形代表切面ABCD与EBFD．在这两个正方形中割出细缝，如图5，并沿BOD将两纸片组合起来．。

当这两张卡片纸互相垂直时，A、B、C、D、E与F6点也就是正八面体的顶点．。

继续完成此模型．剪下第三个正方形代表切面AECF；将正方形沿对角线EF分成两半，再沿着OA与OC割出细缝，如图6；现在将这两片半个正方形附加上去，即完成此模型，再使用胶水或胶带纸固定．。

另一种做模型的方法是使用3个正方形框，重点是强调正八面体的正方形切面(可使用旧的铁丝衣架，且铁丝漆成不同颜色)．用线将各个顶角绑起来，这种模型强调八面体的边．。

将线或松紧带穿入吸管，也可以做出这种强调八面体边的模型(图7)．不过使用吸管时，通常是先做出一个三角形，然后在上面搭出其他三角形，直到模型完成．也可以分别用4根吸管做出3个分开的环，代表切面ABCD、AECF与BEDF，然后将之联接在一起．在最后联接在一起之前，这种模型都不具有内在的刚性．这种方法相当富于启发性．。

由八面体中的一个顶点开始，例如A，可以找到一条路径，走过所有的边而不需重复经过任何边就返回起点，例如：

A→B→E→D→F→B→C→D→A→E→C→F→A。

杜德尼(H．E．Dudney)曾以此为基础设计了一道谜题，他向读者提出挑战，要找出由一个顶点开始究竟有多少条此种路径．路径的数目大得惊人，请你也试着找找看．。

既然有此种路径存在，就表示你能用12根吸管连接成的封闭环做出一个吸管八面体．请试一试．。

如果把吸管八面体置于幕布之前，再用光照射，则会出现各种不同形状的投影，但最令人惊奇的是会出现六边形与其对角线(图8)．这是怎么做到的？

只要在吸管模型的一面加上3根吸管，就可以轻易地做出一个四面体．如果在八面体的各个面间隔地做出此种四面体，结果就是一个较大的四面体．。

另一种观察正八面体与正四面体之间关系的方法是将正四面体的角对称地截去，参见图9．。

如果以正八面体为起点并在其8个面上都加一个四面体，结果将成为一个八角星或是两个互相穿插的正四面体，而两者中间的共同部分就是最初的那个正八面体，参见图10．。

现在仔细观察八角星，你可以发现各角也是正方体的顶点，参见图11；同时，最初的正八面体的顶点也恰好位于正方体各面的中心，参见图12．。

其实，正方体与正八面体之间关系之密切远不只如此．如果以正八面体为起点，将相邻面的中点画线连接，就可以形成正方体，参见图13．因此，我们称正方体与正八面体互为“对偶”(dual)型立体，而且它们具有相同的对称性．正方体的任何对称面也都是正八面体的对称面．同理，旋转对称轴也是一样．同时，无论是正方体还是正八面体，截角到最后的形状都是“方形八面体”(cuboctahedron)，参见图14．。

天然的晶体通常会形成各种形状，例如一般的氯化钠晶体为正方体，明矾晶体为正八面体，辉银矿石的晶体为方形八面体．只要我们了解球体能以各种方式堆叠在一起充填空间，就会觉得晶体形状各异其实并不足以为奇．下列图形显示较常见的几种排列方式及其与各种形状之间的关系，不过要真的了解两者的关系，最好是用小球做出模型．。

在图15与图16中，球在每一层都排成正方形，而在新的一层上也是一样．这称为“正方体填充”(cubical packing)，如图15．如果考虑6个球要触及某一特定的球，参见图16，则那6个球的中心就位于正八面体的顶点．如一层球排成正方形，而新的一层球均位于前一层球形成的凹洞之中，也能显现出正八面体的形状，参见图17．方形八面体可以看成是一层球排成六边形，而新的一层球则位于前一层球形成的各个凹洞中，参见图18．在这种情况下要注意的是在间隔的层之间，球并没有直接上下相连，但是对应着由中间一层的球所形成的凹洞．。

达芬奇英语简介

在晶体结构中，一个原子或离子总是按某种方式与周围的原子或异号离子相邻结合，形成所谓的配位(coordination)关系。后者可以用配位数和配位多面体来具体描述。

一个原子或离子的配位数(coordination number，缩写为CN)是指:晶体结构中，在该原子或离子的周围，与它直接相邻结合的原子个数或所有异号离子的个数。配位多面体(coordination polyhedron)则是指:在晶体结构中，与某一个阳离子(或中心原子)成配位关系而相邻结合的各个阴离子(或周围的原子)，它们的中心连线所构成的几何多面体。阳离子(或中心原子)即位于配位多面体的中心，与它配位的各个阴离子(或配位原子)的中心则位于配位多面体的角顶上。

对于矿物晶体来说，最具有重要意义的是离子化合物晶体中阳离子的配位数。如8.3.2小节中所述，在离子晶格中，通常是阴离子作最紧密堆积，阳离子充填其中的八面体空隙或四面体空隙。显然，在此情况下阳离子的配位数应分别为6和4，相应的配位多面体分别为配位八面体和配位四面体(图8.6C和B)。根据实际资料分析，证实在大多数离子晶格中，阳离子的配位多面体确以八面体配位(CN=6)和四面体配位(CN=4)最为常见。但在阴离子不成最紧密堆积的情况下，也还存在着其他的配位数。

图8.6 阳离子的几种典型的配位型式及其相应的配位多面体。

在阴离子作紧密堆积的情况下，可以从理论上计算出阳离子的配位数与阴、阳离子半径比值之间的关系。例如，当阴离子作最紧密堆积时，假若某个阳离子恰好能无间隙地充填其中的八面体空隙，与周围的6个阴离子均紧密接触，且阴离子本身相互间也都直接接触，那么设阳离子半径为rc，阴离子半径为ra，并过配位八面体的中心作垂直于其任一L4的切面，从中易于看出。

结晶学导论

显然，此0.414即是阳离子无间隙地恰好填满八面体空隙时的阴、阳离子半径之比值，它应当是阴离子作最紧密堆积时，阳离子可能作八面体配位的下限值。当半径比小于此值时，就表明阳离子将不再能无间隙地充填八面体空隙，它不再能同时与周围的所有阴离子都紧密连接。离子的这种位置显然不是达到平衡的位置。因此，这时阳离子实际上只能存在于较八面体空隙小一点的四面体空隙中。当然，此时它通常需要把周围的阴离子撑开一些，从而使阴离子本身相互间不再直接接触。如果半径比继续减小而达到某个一定值时，阳离子不必再将阴离子撑开，即可以无间隙地正好充填于四面体空隙中。那么这个比值便应是阳离子可能作四面体配位的下限值;而上述八面体配位的下限值，同时也就是四面体配位的上限值。如图8.7A，先作一立方体，使其相间的四个角顶A、B、C、D两两连线，所得立方体的6条面对角线即构成一个四面体(此时阳离子中心与立方体中心O重合)，并设立方体之棱长为2，则在图8.7B之(110)切面ABEF上可见。

结晶学导论

图8.7 求解四面体配位时阴、阳离子半径比之下限值的图解A—四面体及其辅助立方体之关系;B—过辅助立方体中心的(110)切面图(罗谷风，2009)。

从而得出四面体配位时阴、阳离子半径比的下限值为0.225。显然，当半径比小于此值时，阳离子只能进入单层阴离子最紧密排列层内的弧线三角形空隙中，0.225即为此三角形配位的上限值，而其下限值应为0.155。若小于此值，阳离子便只能挤开二相邻阴离子而在其间形成2次的哑铃状配位。反之，处于八面体空隙中的阳离子，当rc/ra＞0.414时，它就需要将周围的阴离子撑开些，且随着半径比的增大，阴离子本身相互间的间隙也被撑开得越来越大。当rc/ra=0.732时，阳离子就有可能同时与8个阴离子相邻接触，即采取8次的立方体配位，阴离子则相应地改变堆积方式，成为相当于立方原始格子形成的简单立方堆积(见图8.6D)，其空隙率达到47.64%。这样，又得出了八面体配位的上限值为0.732。至于立方体配位中当阳离子增大到rc=ra时，实际上就成了等大金属原子的最紧密堆积，任一原子的配位数均为12，但因其堆积方式为三层或是两层重复的CCP或HCP而有两种不同的配位多面体，如图8.8所示。两者都由6个正方形面和8个等边三角形面组成，但对应于CCP者其14个面成对平行分布，构成立方体与八面体的聚形———立方-八面体(cuboctahedron);对应于HCP者其7对面相对于最紧密排列面成上下对称分布，构成由两个三方双锥和一个平行双面组成的聚形———六方复七面体(hexagonal diheptahedron)。

图8.8 等大球体最紧密堆积的两种基本型式和对应的12次配位多面体。

综上所述，在阴离子成紧密堆积条件下，可将阳离子配位多面体与阴、阳离子半径比值间的关系简列成如表8.2。

表8.2 球体紧密堆积中配位多面体与阴、阳离子半径比的几何关系。

①写于括弧内的实际半径比不符合理论计算所得出的范围。

应当强调，表8.2中的结果和之前的讨论都是以阴离子成紧密堆积而从纯几何学角度作出的。因此，它们除不适用于原子晶格和分子晶格外，在离子晶格中实际上也存在不少例外，包括:①可出现5、7、9、10、11等配位数。②配位数相同而配位多面体可能不同。例如CN=6和4者，分别还可有配位三方柱和配位正方形;CN=5的有配位四方锥和配位三方双锥;等等。③配位多面体形状存在畸变的情况相当常见。④半径比不等于1的大阳离子也可能具有12次配位。例如钙钛矿中的Ca2+、黑云母中的K+等。⑤大多数CN=4的配位多面体，其半径比经常小于理论下限值;CN=3者这一情况更为突出。⑥CN=2的在矿物中仅有赤铜矿一个实例(表8.2)。⑦过渡型离子的配位常可出现明显违反半径比关系的相反倒置现象。

导致存在以上情况的原因主要有:晶体的对称性较低，致使阴离子的堆积或多或少偏离紧密堆积;过渡型离子所形成的离子键中含有较高的共价键成分，或者本身就是多键型晶格;存在强烈的离子极化;阳离子的外层电子构型有所不同;晶格保持局域静电平衡的需要;过渡型离子受晶体场作用的影响;等等。以上因素往往不是各自孤立的，许多现象常是多种因素共同作用的结果。但是尽管有许多例外，表8.2的关系就其总体而言，特别是对于最常见的八面体配位和四面体配位，是很有用的。同时，对于金属晶格更是相当吻合的，绝大多数金属单质晶体都呈CCP、HCP或立方体心紧密堆积，具有最高的12次或较高的8次配位。

此外，以上讨论中，配位多面体都是按几何上的正多面体来处理的，而实际上往往可有程度不同的畸变。但这并不影响有关结论的基本正确性，并仍以正多面体的术语来称呼它们。

达芬奇评价

达芬奇，意大利学者、艺术家。也是欧洲文艺复兴时期的著名人物，现代学者称他为“文艺复兴时期最完美的代表”，下面是我给大家整理的达芬奇英语简介，供大家参阅!。

达芬奇简介 Leonardo di Piero da Vinci (Italian name: Leonardo di ser Piero da Vinci, Julian calendar April 15, 1452 (Gregorian calendar April 23) - May 2, 1519 , At the age of 67), the Italian scholar, the artist.。

European Renaissance genius scientist, inventor, painter. Modern scholars call him the "perfect representative of the Renaissance", is the unique history of human history, his greatest achievement is painting, his masterpiece "Mona Lisa", "the last dinner", "rock Our Lady "and other works, embodies his superb artistic attainments. He believes that the most beautiful natural object of study is the human body, the human body is the wonderful works of nature, the painter should be the core of the object of painting.。

He is a deep thinking, knowledgeable, versatile painter, astronomer, inventor, construction engineer. He is also good at sculpture, music, invention, architecture, familiar with mathematics, physics, physics, astronomy, geology and other disciplines, both versatile, and hard work, save the manuscript about 6000 pages. All his research results are kept in his manuscript, Einstein believes that Leonardo da Vinci's scientific research if published at the time, then the technology can be 30-50 years ahead of time.。

Da Vinci juvenile has been revealed artistic talent, about 15 years old to Florence apprentice learning, grow into a scientific literacy painter, sculptor. And became a military engineer and architect in 1482 after graduation to Milan graduated from the Italian Institute of Technology to become a famous Italian architect, artist, in the aristocratic palace in the creation and research activities, since 1513 drifted in Rome and Florence and other places. 1516 living in France, May 15, 1519 died. Asteroid 3000 was named "Leonardo". The most famous work is "Mona Lisa" is now one of the three treasures of the Louvre in Paris.。

达芬奇人物生平 On April 15 of 1452 (the Julian calendar), Leonardo da Vinci was born three hours later in the night "near the valleys of the Tuscan hills of the Finch, near the valleys flowing through the Arno River, The family of the Republic of Florence under the reign of the Dodge family, the father of Leonardo da Vinci, called Pierre Pierre Da Vinci, is the legal notary of Florence and is so rich in his mother. • Finch is their illegitimate son. Da Vinci does not have a real name, his full name "Lionardo di ser Piero da Vinci" means: "Vinci town Moses Piero son - Leo Naduo. "His name" ser "only shows that his father is a gentleman.。

We know very little about Leonardo da Vinci's childhood. He lived with his mother in the village of Finch before the age of five, and after 1457 and his father, grandparents, uncle Francesco lived in the town of Finch. His father and the 16-year-old girl named Albiera got married. She loves da Vinci, but dies early Leonardo da Vinci was sixteen years old when his father married the two-year-old Francesca Lanfredini. Until his third and fourth marriage, Leonardo da Vinci's father had a legitimate heir.。

Da Vinci did not formally learn Latin, geometric and math.。

Later, Da Vinci recorded his childhood two small accidents. Once there was a bird hovering over his cradle, its tail's feathers swept his face. Another time, he found a cave in the mountains to explore, although the fear will be hidden inside the monster, but in the curiosity of the final drive or go to find out. The。

Leonardo da Vinci's childhood has become a historical problem. In the sixteenth century, the Renaissance biographer and painter Joel Qiao Vasari, described a local farmer who had made a shield, and asked Leonardo da Vinci's father, Pierre, to paint on the shield. Son Da Vinci painted a monster that spit on the flames. This painting is lifelike, it is very scary, Piero put it to the Florentine art intermediary, art intermediary and the painting sold to the Duke of Milan. Then Piero was using the transfer of the painting to make money to the farmer to buy a new shield, painted above a red arrows through the arrow.。

We are more familiar with the image of Da Vinci basically from his famous self-portrait, so a mention of Leonardo, we always think of a philosopher-like wise elders. In fact, Leonardo da Vinci young age, but Italy, Florence, famous man! (But because he was not interested in women, so then he was gay rumors flying)!。

His teacher Verroeo sculpture of the handsome extraordinary bronze David is said to be the young Da Vinci as a model. Da Vinci's artistic career developed most smoothly in the period 1482 - 1499 Milan. Leonardo da Vinci's lyre played well, he first appeared as a musician rather than a painter or inventor in Milan. During this period of his paintings are not many, but its unparalleled talent is very popular with Milan Grand Duchess of Dovico Svolza.。

1499 years to avoid the war Da Vinci in Mantua and Venice and other places to travel and conduct some scientific research.。

In 1500 Leonardo da Vinci returned to Florence and began to create "Mona Lisa". "Mona Lisa" using a perspective method and other painting methods. After that, Leonardo da Vinci went to Milan and continued to serve the Milan court.。

1513 moved to Rome, Rome for Leonardo is not very pleasant place. Where he made a short stay, saw Michelangelo and other artists at the time in Rome, but did not reveal any of his artistic genius. Where he is basically studying some similar to the magic of the trick, so that the Romans when he is a shaman class of people.。

In 1516 Da Vinci went to France and finally settled in Amboise.。

In his later years rarely painting, devote themselves to scientific research, leaving a lot of notes handed down manuscripts, content from physics, mathematics to biological anatomy, almost all-encompassing. There are not many works of his life, but pieces are immortal. His works have a distinct personal style and are good at combining art creation with scientific inquiry, which is unique in the history of world art. The academic community generally divides its creative activities into two stages of early and peak.。

达芬奇科学 Renaissance humanism (Renaissance humanism) between science and art is not mutually exclusive extreme phenomenon. Da Vinci studied science and engineering as his artwork was memorable and outstanding. About 13,000 pages of notes and paintings in the manuscript are all records of mixed art and science. These records are Da Vinci in the European journey through the observation of the surrounding and continue to create.。

He wrote his mirror all his life. For the left hand writer, the quill pen is pulled from right to left to write than from left to right to write easy to write, and will not just write the word paste. So his diary is a mirror word.。

In science, Da Vinci is a huge observer, can be a very fine description of the way that a phenomenon. But not through the theory and experiment to verify. Because of the lack of formal education in Latin and mathematics, most of the scholars at the same time did not notice Da Vinci in the field of science. And Da Vinci rely on self-learning to understand Latin. It has also been said that Leonardo da Vinci intends to publish a series of papers containing various subjects, but never achieved.。

The rhombicuboctahedron, painted by Leonardo da Vinci, appeared in the "Divina Proportione" of Luca Pacioli in 1509.。

Leonardo da Vinci began to understand human anatomy when he studied with Andrea del Verrocchio. At that time Verrochio insisted that all disciples learn anatomy. When Da Vinci became a successful artist, he was given permission to dissect the human body at the hospital Santa Maria Nuova hospital in Florence. After which he worked at the hospital Maggiore and the hospital of Santo Spirito (the first Italian local hospital). In 1510 to 1511, he worked with Doctor Marcantonio della Torre; 1481 - 1511). Within 30 years, Da Vinci dissected a total of 30 different sexes of the human body. When working with Dr. Thor, Da Vinci was ready to publish anatomical works and painted more than 200 paintings.。

However, his book until the 1680 AD (died in 161 years) to "painting" in the name of the publication. In addition to the human body, Leonardo da Vinci also dissected cattle, poultry, monkeys, bears, frogs as anatomical structure.。

Leonardo da Vinci painted many human skeletal iliacs, and he was also the first person to describe the spine double S type. He also studied the pelvic and sacral inclination as well as emphasizing the sacral not only a single form, but also the five vertebrae. Leonardo da Vinci is also able to show the shape of the skull and the cross-section of the brain (cross section, longitudinal section, tangent plane). He painted many maps including the lungs, mesenteric, urinary tracts, sexual organs and even sexual intercourse. He is the first to draw the fetus in the womb (he wants to understand the "birth miracle") is also the first to draw the abdominal cavity in the appendix. In addition Da Vinci often depicts the muscles and tendons of the neck and shoulders. In 2005, Leonardo da Vinci's paintings inspired a British heart surgeon to develop a new way to repair damaged hearts.。

He can be said that the local anatomy of the master, not only the study of human anatomy outstanding performance, but also in the study of other animal anatomy when the same outstanding performance. Leonardo da Vinci not only concerned with the body structure, but also concerned about the physiological function, which makes him become anatomists and physiologists. He actively looks for the appearance of obvious physical defects as a model in order to draw a funny painting highlights the significance of physiological function. It is worth mentioning that Da Vinci can write one hand at the same time.。

Based on human anatomical research also makes Da Vinci design the first robot in history. This design, called Leonardo's robot, may have been completed in 1495, but it was not until the 1950s. I wonder if this design is to be made into an entity.。

达芬奇的发明有那些？

达芬奇是欧洲文艺复兴时期的天才科学家、发明家、建筑工程师、画家。他还擅长雕塑、音乐、发明、建筑，通晓数学、生理、物理、天文、地质等学科，既多才多艺，又勤奋多产，保存下来的手稿大约有6000页。现代学者称他为“文艺复兴时期最完美的代表”，是人类历史上绝无仅有的全才，他最大的成就是绘画，他的杰作《蒙娜丽莎》、《最后的晚餐》、《岩间圣母》等作品，体现了他精湛的艺术造诣。

艺术创作

早期创作。当他在作坊学艺时 ，就表现出非凡的绘画天才。现存他最早的作品《受胎告知》是达芬奇在没有老师的指导下，独立完成的一件作品。。稍后创作的《吉内薇拉·班琪》，一反15世纪艺术追求线条分明的传统，以逆光夕照的色调渲染他所倡导的透视效果。1481年创作的又译《三王来拜》是标志其艺术风格达到成熟期的作品。

盛期创作。1482年达芬奇来到米兰，应圣弗朗切斯教堂的邀请绘制祭坛画《岩间圣母》。这幅画现藏于卢浮宫。《最后的晚餐》是他在这一时期的创作中最负盛名之作。5世纪末期，为米兰公爵卢多维科·斯福尔扎的情妇切奇利娅·加莱拉尼创作了《抱银鼠的女子》。《抱银鼠的女子》是达·芬奇杰出的肖像画作品，体现了深厚的写实功力。

科学。达芬奇钻研科学、工程领域就像他的艺术作品般令人难忘与突出。手稿中约13,000页的笔记与绘画全是混合艺术与科学所组成的纪录。这些纪录是达芬奇在欧陆行程中透过观察周遭并不断的创作而来。在科学上，达芬奇是一个巨细靡遗的观察家，能以极精细的描述手法表示一个现象。

达芬奇所绘的菱方八面体(rhombicuboctahedron)，1509年出现在卢卡·帕西欧里(Luca Pacioli)的《神圣比例》(Divina Proportione) 中。

医学。像15世纪的其他学者一样，达芬奇也通过阅读古代典籍来支持自己的研究。为获取解剖学知识，他阅读了古希腊医生盖伦（Galen）的著作。当达芬奇成为成功的艺术家时，得到于佛罗伦斯圣玛丽亚纽瓦医院(hospital Santa Maria Nuova)解剖人体的许可。30年内，达芬奇共解剖了30具不同性别年龄的人体。当与托尔医生共同工作时，达芬奇准备出版解剖学理作品并制绘了超过200篇画作。他可说是局部解剖图宗师，不仅研究人体解剖表现杰出，而且在涉猎其他动物解剖时一样表现出色。

工程。对建筑与工程学的兴趣将他引向了古罗马建筑师维特鲁威的著作，后者的论文在十五世纪早些时候启发了阿尔贝蒂。由于着迷飞行现象，达芬奇作了鸟类飞行的详细研究，同时策划了数部飞行机器，包括了直升机设计图以及轻型滑翔翼。达芬奇曾任军事工程师，笔记中也包含了数种军事机械的设计：机关枪、人力或以马拉动的武装坦克车、子母弹、军用降落伞、含呼吸软管以猪皮制成的潜水装等等。

谢谢阅读！

艺术为生活添色彩。公众号【瑞亚子】

原文地址：http://www.qianchusai.com/cuboctahedron.html