12.7 发射

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  Tessellations
  ::发射

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  A tessellation is a tiling over a plane with one or more figures such that the figures fill the plane with no overlaps and no gaps. You have probably seen tessellations before. Examples of a tessellation are: a tile floor, a brick or block wall, a checker or chess board, and a fabric pattern. The following pictures are also examples of tessellations.
  ::通融是平面上的一个砖块,有一个或多个数字,使数字填满平面,没有重叠,没有空白。您可能已经见过多次通融,例如:瓷地板、砖墙或砖墙、棋盘或棋盘以及布局。以下图片也是通融的例子。

  

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  Notice the hexagon (cubes, first tessellation) and the quadrilaterals fit together perfectly. If we keep adding more, they will entirely cover the plane with no gaps or overlaps.
  ::注意六边形( 立方体, 第一个熔化) 和四边形完美地组合在一起。 如果我们继续添加更多, 它们将完全覆盖平面, 没有空白或重叠 。

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  We are only going to worry about tessellating regular polygons. To tessellate a shape, it must be able to exactly surround a point , or the sum of the angles around each point in a tessellation must be $\begin{align*}360^\circ\end{align*}$ . The only regular polygons with this feature are , squares, and regular hexagons.
  ::我们只会担心连接正态多边形。 要将形状变形, 它必须能够完全环绕某个点, 或星系中每个点周围角度的总和必须是 360 。 只有具有此特性的正态多边形是, 方形和正态六边形 。

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  What if you were given a hexagon and asked to tile it over a plane such that it would fill the plane with no overlaps and no gaps?
  ::如果有人给了你一个六边形,要求你用平面把它堆在飞机上,以便填满飞机,没有重叠,没有空隙,那又如何?

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  Examples
  ::实例

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  Example 1
  ::例1

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  How many regular hexagons will fit around one point?
  ::有多少普通的六边形将适合一个点左右?

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  First, recall how many degrees are in a circle , and then figure out how many degrees are in each angle of a regular hexagon. There are $\begin{align*}360^\circ\end{align*}$ in a circle and $\begin{align*}120^\circ\end{align*}$ in each interior angle of a hexagon, so $\begin{align*}\frac{360}{120}=3\end{align*}$ hexagons will fit around one point.
  ::首先,请记住一个圆圈中有多少度, 然后找出一个普通六边形的每个角度中有多少度。 在一个圆圈里有360°, 在一个六边形的每个内角里有120°, 所以 360120=3六边形将适合一个点左右 。

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  Example 2
  ::例2

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  Does a regular octagon tessellate?
  ::普通的八边方星座板吗?

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  First, recall that there are $\begin{align*}1080^\circ\end{align*}$ in a pentagon . Each angle in a regular pentagon is $\begin{align*}1080^\circ \div 8 = 135^\circ\end{align*}$ . From this, we know that a regular octagon will not tessellate by itself because $\begin{align*}135^\circ\end{align*}$ does not go evenly into $\begin{align*}360^\circ\end{align*}$ .
  ::首先, 请记住在五角形中存在 1080 。 普通五角形中每个角度是 1080 8=135 。 我们从中知道, 普通八边形不会自动叠叠, 因为 135 不均匀地切入 360 。

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  Example 3
  ::例3

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  Draw a tessellation of equilateral triangles.
  ::绘制等边三角形的星系。

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  In an equilateral triangle each angle is $\begin{align*}60^\circ\end{align*}$ . Therefore, six triangles will perfectly fit around each point.
  ::在等边三角形中,每个角度是60 。 因此, 六个三角形将完全适合每个点 。

  

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  Extending the pattern, we have:
  ::在扩展模式时,我们有:

  

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  Example 4
  ::例4

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  Does a regular pentagon tessellate?
  ::普通的五金色套件吗?

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  First, recall that there are $\begin{align*}540^\circ\end{align*}$ in a pentagon. Each angle in a regular pentagon is $\begin{align*}540^\circ \div 5 = 108^\circ\end{align*}$ . From this, we know that a regular pentagon will not tessellate by itself because $\begin{align*}108^\circ\end{align*}$ times 2 or 3 does not equal $\begin{align*}360^\circ\end{align*}$ .
  ::首先,请记住,在五角形中,有540。在普通五角形中,每个角度是 5405=108。从这一点上,我们知道,普通五角形不会自行变形,因为108乘以2或3不等于360。

  

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  Example 5
  ::例5

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  How many squares will fit around one point?
  ::多少平方块将适合一个点左右?

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  First, recall how many degrees are in a circle, and then figure out how many degrees are in each angle of a square . There are $\begin{align*}360^\circ\end{align*}$ in a circle and $\begin{align*}90^\circ\end{align*}$ in each interior angle of a square, so $\begin{align*}\frac{360}{90}=4\end{align*}$ squares will fit around one point.
  ::首先, 记得一个圆圈中有多少度, 然后找出一个平方的每个角度中有多少度。 一个圆圈里有 360 °, 一个平方的每个内部角里有 90 °, 所以 36090= 4 的方形将适合一个点左右 。

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  Review
  ::回顾

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  1. Tessellate a square. Add color to your design.
     ::泰塞拉特方块 给设计增加颜色
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  2. What is an example of a tessellated square in real life?
     ::真实生活中的书信广场是什么样?
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  3. Tessellate a regular hexagon. Add color to your design.
     ::插入普通六边形,在设计中添加颜色。
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  4. You can also tessellate two regular polygons together. Try tessellating a regular hexagon and an equilateral triangle. First, determine how many of each fit around a point and then repeat the pattern. Add color to your design.
     ::您也可以将两个正则多边形一起捆绑在一起。 尝试将一个正则六边形和一个等边三角形串联起来。 首先, 确定每个三角形中有多少个适合一个点, 然后重复这个图案。 将颜色添加到设计中 。
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  5. Does a regular dodecagon (12-sided shape) tessellate? Why of why not?
     ::普通的十二进制( 12 面形) etselolate 吗? 为什么不呢 ?
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  6. Does a kite tessellate? Draw an example of how it might be possible.
     ::是否有风筝板板? 请举一个例子说明它是如何可能的。

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  Do the following figures tessellate?
  ::是否使用以下数字 etselplate ?

  7. 
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  13. 
  14. 

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  Review (Answers)
  ::回顾(答复)

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  Click to see the answer key or go to the Table of Contents and click on the Answer Key under the 'Other Versions' option.
  ::单击可查看答题键, 或转到目录中, 单击“ 其他版本” 选项下的答题键 。