1 Introduction
RegularNGons
and its numerical counterpart. Finally, Sect. 5 depicts some future ideas.2 Mathematical Background
2.1 Constructibility
2.2 An Algebraic Formula for the Vertices
n | Minimal polynomial of \(\cos \frac{2\pi }{n}\) |
---|---|
1 | \(x-1\) |
2 | \(x+1\) |
3 | \(2x+1\) |
4 | x |
5 | \(4x^2+2x-1\) |
6 | \(2x-1\) |
7 | \(8x^3+4x^2-4x-1\) |
8 | \(2x^2-1\) |
9 | \(8x^3-6x+1\) |
10 | \(4x^2-2x-1\) |
11 | \(32x^5+16x^4-32x^3-12x^2+6x+1\) |
12 | \(4x^2-3\) |
13 | \(64x^6+32x^5-80x^4-32x^3+24x^2+6x-1\) |
14 | \(8x^3-4x^2-4x+1\) |
15 | \(16x^4-8x^3-16x^2+8x+1\) |
16 | \(8x^4-8x^2+1\) |
17 | \(256x^8+128x^7-448x^6-192x^5+240x^4+80x^3-40x^2-8x+1\) |
3 Manual Results on Regular 5- and 11-gons
3.1 Some Properties of a Regular Pentagon
3.2 Regular Star-Polygons
3.3 Lengths in a Regular 11-gon
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\(b=c\),
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\(d=e\),
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Triangles CLM and CON are congruent,
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\(a=l\) (that is, \(AB=DL\)).
4 Automated Discovery of Theorems
4.1 A Bijective Mapping?
4.1.1 Example
4.2 A Symbolic Implementation
RegularNGons
.-
\(n=\ldots \) defines the number of vertices in the regular polygon.
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s and e define the starting and ending cases (both are non-negative integers, less than the formula (4.1)).
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By adding \(m=\ldots \) or \(M=\ldots \) the minimal and maximal degrees of outputs can be controlled, respectively. By default \(m=1\) and \(M=2\), that is, either linear results or quadratic surds are mined.
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The parameter u will force searching for results given as parameters. For example, \(u=2\) considers only the outputs that are of \(q=2\).
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The option \(S=0\) tries to avoid checking cases that were already checked in a symmetrically equivalent position. When this is set, only the \(A=0\), \(B\le n/2\) cases will be checked. (The software tool uses the indices of the points, that is, 0 stands for \(P_0\), 1 for \(P_1\), and so on.)
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When using \(f=1\), once a length is found, no more results will be printed that have the same length.
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The user may request to find lengths that are close to a given decimal number, but they are just approximately the same. The parameter \(a=\ldots \) is to be set to the sought decimal. (See Sect. 4.6 for some examples.) By using the parameter E an error limit can be defined.
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The option \(z=\ldots \) allows saving and retreiving results later by using an external server.
RegularNGons
is the following, when using inputs \(n=7\), \(S=0\) and \(f=1\): This result will be recalled later in Theorem 4.15.
4.3 A Numerical Implementation in GeoGebra
4.4 Some Results
RegularNGons
.-
Points of the first kind of a regular n-gon are its vertices. We denote this set by \({{\mathcal {P}}}_1\).
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Segments of the first kind of a regular n-gon are its sides and diagonals. We denote this set by \({{\mathcal {S}}}_1\).
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Points of the k-th kind of a regular n-gon are the intersection points of its segments of the \((k-1)\)-th kind. We denote this set by \({{\mathcal {P}}}_k\).
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Segments of the k-th kind of a regular n-gon are the segments defined by its points of the (k)-th kind. We denote this set by \({{\mathcal {S}}}_k\).
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\(|{{\mathcal {P}}}_1|=n\).
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\(|{{\mathcal {S}}}_1|=\left( {\begin{array}{c}|{{\mathcal {P}}}_1|\\ 2\end{array}}\right) \).
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\(|{{\mathcal {P}}}_k|\le \left( {\begin{array}{c}|{{\mathcal {S}}}_{k-1}|\\ 2\end{array}}\right) \).
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\(|{{\mathcal {S}}}_k|=\left( {\begin{array}{c}|{{\mathcal {P}}}_{k}|\\ 2\end{array}}\right) \).
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\({{\mathcal {P}}}_1\subseteq {{\mathcal {P}}}_2\subseteq {{\mathcal {P}}}_3\subseteq \cdots \)
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\({{\mathcal {S}}}_1\subseteq {{\mathcal {S}}}_2\subseteq {{\mathcal {S}}}_3\subseteq \cdots \)
4.5 Correspondence Between Elimination and the Minimal Polynomial of the Expected Length of RS
RegularNGons
(note that \(\varphi (24)=8\)): \(A=0\), \(B=1\), \(C=0\), \(D=2\), \(E=0\), \(F=1\), \(G=2\), \(H=8\),-
\(\{13/4\}\) and \(\{13/6\}\),
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\(\{13/2\}\) and \(\{13/3\}\),
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\(\{13\}\) and \(\{13/5\}\),
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\(\{13\}\), \(\{13/3\}\) and \(\{13/4\}\),
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\(\{13/2\}\), \(\{13/5\}\) and \(\{13/6\}\),
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In a regular heptagon the only rational lengths in \({\mathcal {S}}_2\) are 1 and 2, and the only quadratic surd is \(\sqrt{2}\).
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In a regular nonagon the only rational lengths in \({\mathcal {S}}_2\) are 1, 2 and 3, and the only quadratic surds are \(\sqrt{3}\) and \(\sqrt{7}\).
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In a regular 11-gon the only rational lengths in \({\mathcal {S}}_2\) are 1, 2, and the only quadratic surd is \(\sqrt{3}\).
4.6 Approximate Results in Regular 11-gons
RegularNGons
, one may obtain some “almost”-results that can be interesting when creating tricky problem assignments. Here some results are listed—most of them involve regular star-polygons.4.7 Approximating \(\pi \)
n | Numerically | Symbolically | “Minimal polynomial” | Example case |
---|---|---|---|---|
3 | 1 | 1 | \(q-1\) | 0 |
4 | 1.41... | \(\sqrt{2}\) | \(q^2-2\) | 5 |
5 | 3.07... | \(\sqrt{2\sqrt{5}+5}\) | \(q^4-10q^2+5\) | 33 |
6 | 3.12... | \(\frac{\sqrt{39}}{2}\) | \(q^2-\frac{39}{4}\) | 936 |
7 | 3.16... | \(q^6-24q^4+143q^2-29\) | 2496 | |
8 | 3.13... | \(\sqrt{2\sqrt{2}+7}\) | \(q^4-14q^2+41\) | 200 |
9 | 3.13... | \(q^6-30q^4+237q^2-379\) | 67,311 | |
10 | 3.1413... | \(\sqrt{\frac{2\sqrt{5}+35}{4}}\) | \(q^4-\frac{35}{2}q^2+\frac{1205}{16}\) | 19,113 |
11 | 3.1411... | \(q^{10}-34q^8+381q^6-1669q^4+2687q^2-1277\) | 29,802 |
4.8 Other Examples
RegularNGons
can be launched on-line at http://prover-test.geogebra.org/~kovzol/RegularNGons/. An example run can be started to request solving the case \(n=5\) by invoking the URL http://prover-test.geogebra.org/~kovzol/RegularNGons/?n=5.