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| \PMlinkescapetext{This page is under construction.} |
\PMlinkescapetext{This page is under construction.} |
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| Some preliminary definitions are \PMlinkescapetext{necessary}: |
Some preliminary definitions are \PMlinkescapetext{necessary}: |
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| \begin{itemize} |
\begin{itemize} |
| \item A \emph{compass} is a tool which can be used for drawing circles, or arcs thereof, whose radii are sufficiently small, and for measuring lengths. |
\item A \emph{compass} is a tool which can be used for drawing circles, or arcs thereof, whose radii are sufficiently small, and for measuring lengths. |
| \item A \emph{straightedge} is a tool which can be used for drawing lines, or segments thereof. |
\item A \emph{straightedge} is a tool which can be used for drawing lines, or segments thereof. |
| \end{itemize} |
\end{itemize} |
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| Some people use the word \emph{ruler} to refer to the tool that is called a straightedge here. This can cause some confusion, however, because outside of mathematics, a ruler is used to measure any \PMlinkname{length}{BasicLength} desired. This is \emph{not} a permissible tool for compass and straightedge constructions. |
Some people use the word \emph{ruler} to refer to the tool that is called a straightedge here. This can cause some confusion, however, because outside of mathematics, a ruler is used to measure any \PMlinkname{length}{BasicLength} desired. This is \emph{not} a permissible tool for compass and straightedge constructions. |
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| With these preliminaries out of the way, we can now proceed to the main definition. |
With these preliminaries out of the way, we can now proceed to the main definition. |
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A \emph{compass and straightedge construction} is the provable creation of a geometric figure on the Euclidean plane (or complex plane) such that the figure is created using only a compass, a straightedge, and specified geometric figures.
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A \emph{compass and straightedge construction} is the e creation of a geometric figure on the Euclidean plane (or complex plane) such that the figure is created using only a compass, a straightedge, and specified geometric figures.
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Typically, if no preexisting geometric figure is specified, the tacit assumption is that one can use a line segment of length $1$. Moreover, in such instances, one can specify what length represents $1$, but it must remain constant throughout the construction..
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Typically, if no preexisting geometric figure is specified, the tacit assumption is that one can use a line segment of length $1$. Moreover, in such instances, one can specify what length represents $1$, but it must remain constant throughout the construction. One has to be very careful with this usage. For example, the phrase ``A $20^{\circ}$ angle is not constructible with compass and straightedge'' refers to the fact that a compass and straightedge construction of a $20^{\circ}$ angle is not possible using the tacit assumption as described above.
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A geometric figure is \emph{constructible} if it can be made from a compass and straightedge construction.
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Before moving on to examples, some more definitions need to be made.
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| One has to be very careful with the terminology associated with compass and straightedge constructions. For example, the phrase ``A $20^{\circ}$ angle is not constructible with compass and straightedge'' refers to the fact that a compass and straightedge construction of a $20^{\circ}$ angle is not possible using the tacit assumption as described above. As another example, the following is an erroneous argument regarding compass and straightedge constructions: |
constructible length? |
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| \begin{quote} |
constructible point? |
| A line segment of length $\pi$ is constructible because, given a line segment of length $1$, I can extend it as a ray. Then I can measure the distance between the two endpoints with my compass. After that, I can open the compass $\pi$ times wider. Finally, I can mark that distance on the ray from one of the endpoints of the original line segment. |
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| \end{quote} |
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| The above argument is one of many reasons why the word \emph{provable} appears in the definition. One can open the compass however wide one wants, and one can mark arcs and points however one wants, but the construction is invalid unless one can prove indisputably that the construction really is what is stated. In the above example, one cannot prove that the compass was opened \emph{exactly} $\pi$ times wider than the length of the original line segment. |
constructible line segment? |
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| Chi has asked for the following to appear in this entry if possible: |
Chi has asked for the following to appear in this entry if possible: |
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| \begin{quote} |
\begin{quote} |
| Other things that might be interesting to add under the ``ruler and compass construction'' would be... |
Other things that might be interesting to add under the ``ruler and compass construction'' would be... |
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| \begin{enumerate} |
\begin{enumerate} |
| \item What if we just use compass alone? |
\item What if we just use compass alone? |
| \item What if we just use ruler alone? |
\item What if we just use ruler alone? |
| \item What if we switch one of the tools with another one, say that can generate pi? |
\item What if we switch one of the tools with another one, say that can generate pi? |
| \item What if we add a third tool to the existing two? |
\item What if we add a third tool to the existing two? |
| \end{enumerate} |
\end{enumerate} |
| \end{quote} |
\end{quote} |
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| I want to address concerns raised by Tom in a correction he filed to me. My response to him was: |
I want to address concerns raised by Tom in a correction he filed to me. My response to him was: |
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| \begin{quote} |
\begin{quote} |
| I have elaborated on the rules for these constructions in the entry ``Euclidean field'', which is a more advanced topic and where such technicalities would be more appreciated. I doubt that high school students would be able to understand a discussion such as what we are talking about in this correction. On the other hand, for the benefit of those who are reading this entry and are interested in the technicalities, I will provide an obvious link to ``Euclidean field''. |
I have elaborated on the rules for these constructions in the entry ``Euclidean field'', which is a more advanced topic and where such technicalities would be more appreciated. I doubt that high school students would be able to understand a discussion such as what we are talking about in this correction. On the other hand, for the benefit of those who are reading this entry and are interested in the technicalities, I will provide an obvious link to ``Euclidean field''. |
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| Now to address your questions: |
Now to address your questions: |
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| \begin{itemize} |
\begin{itemize} |
| \item Is the compass collapsible or not? |
\item Is the compass collapsible or not? |
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| I have added a mention in the entry ``Euclidean field'' that it does not matter whether one uses a collapsible compass or modern-day compass. This is a result with which most (if not all) geometers are very familiar, and I can easily supply a proof upon request. If you really want me to add a proof of this fact to PM, please file a request. |
I have added a mention in the entry ``Euclidean field'' that it does not matter whether one uses a collapsible compass or modern-day compass. This is a result with which most (if not all) geometers are very familiar, and I can easily supply a proof upon request. If you really want me to add a proof of this fact to PM, please file a request. |
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| If I were forced to stipulate which compass is to be used, I would choose the modern-day one, simply because it is more accessible (I do not know anyone who owns a collapsible compass) and it makes the constructions easier. |
If I were forced to stipulate which compass is to be used, I would choose the modern-day one, simply because it is more accessible (I do not know anyone who owns a collapsible compass) and it makes the constructions easier. |
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| \item Can a point be picked at random? |
\item Can a point be picked at random? |
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| *Yes.* I have not yet stipulated this in the ``Euclidean field'' entry. If you want me to, please contact me (filing a correction to ``Euclidean field'' is not the way to do this, as it is CWoo's entry) and let me know. |
*Yes.* I have not yet stipulated this in the ``Euclidean field'' entry. If you want me to, please contact me (filing a correction to ``Euclidean field'' is not the way to do this, as it is CWoo's entry) and let me know. |
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| \item If so, I can get lucky and construct pi! |
\item If so, I can get lucky and construct pi! |
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| Two notes on this: Even if you constructed a line segment of length pi due to choosing a point at random, you would not be able to *prove* (using compass and straightedge) that you have constructed pi. Also, with regards to \PMlinkname{measure theory}{Measure}, the probability of constructing pi at random is zero. On the other hand, constructible numbers and angle measures are according to a specific set of rules as indicated in the entry ``Euclidean field''. Thus, they are *not* at random like constructing pi by chance would be. Again by the entry ``Euclidean field'', the probability of constructing a constructible number is 1 since their construction is guaranteed. Long story made short: Even though you might be able to construct a line segment of length pi by chance, it is still not considered constructible by definition. |
Two notes on this: Even if you constructed a line segment of length pi due to choosing a point at random, you would not be able to *prove* (using compass and straightedge) that you have constructed pi. Also, with regards to \PMlinkname{measure theory}{Measure}, the probability of constructing pi at random is zero. On the other hand, constructible numbers and angle measures are according to a specific set of rules as indicated in the entry ``Euclidean field''. Thus, they are *not* at random like constructing pi by chance would be. Again by the entry ``Euclidean field'', the probability of constructing a constructible number is 1 since their construction is guaranteed. Long story made short: Even though you might be able to construct a line segment of length pi by chance, it is still not considered constructible by definition. |
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| Anyways, I think that the above paragraph is *way* too technical for entries that are supposed to be suitable for high school students!!! I am hesitant to even add this to ``Euclidean field'', though I may consider doing so if you contact me personally and let me know. |
Anyways, I think that the above paragraph is *way* too technical for entries that are supposed to be suitable for high school students!!! I am hesitant to even add this to ``Euclidean field'', though I may consider doing so if you contact me personally and let me know. |
| \end{itemize} |
\end{itemize} |
| \end{quote} |
\end{quote} |
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| Compass and straightedge constructions are of historical significance. The ancient Greeks are the most well-known civilization for investigating these constructions on an elementary level. It should be pointed out that the compasses that they used were \emph{collapsible}. That is, you could open the compass and draw an arc, but immediately after you removed a point of the compass from the plane where you drew the arc, the compass would close completely. It turns out that whether a collapsible compass or a modern-day compass is used to perform these constructions makes no difference. More precisely put, the field of constructible numbers when one uses a collapsible compass and the field of constructible numbers when one uses a modern-day compass are equal. |
Compass and straightedge constructions are of historical significance. The ancient Greeks are the most well-known civilization for investigating these constructions on an elementary level. It should be pointed out that the compasses that they used were \emph{collapsible}. That is, you could open the compass and draw an arc, but immediately after you removed a point of the compass from the plane where you drew the arc, the compass would close completely. It turns out that whether a collapsible compass or a modern-day compass is used to perform these constructions makes no difference. More precisely put, the field of constructible numbers when one uses a collapsible compass and the field of constructible numbers when one uses a modern-day compass are equal. |
