In Chapter 2, we explored properties of convex polygons. In this section, we will create regular concave polygons, called star polygons. In addition to studying the geometrical properties of star polygons, we will observe how the relationship between a pair of numbers determines the shape of the star polygon associated with the number pair.
Subsection6.1.1What is a Star Polygon?
When asked to draw a star, most people will draw a five point star polygon without lifting their pencil even though they are unfamiliar with the terminology. We will generalize this to create star polygons with different number of points. We will also see that our procedure will sometimes give us a regular polygon or no polygon at all.
To ensure that the star polygons we create are regular, we begin with equally spaced points around a circle as shown in Figure 6.1.1.(a). We draw segments from point 1 to point 3, from point 3 to point 5, from point 5 to point 2, from point 2 to point 4, and then returning to point 1 to get the star polygon in Figure 6.1.1.(b). Note that we are connecting every second point and continuing until we return to our starting point. There was no need to lift our pencil in this construction.
(a)Five points on a circle.
(b)The \(\{_{2}^{5}\}\) star polygon
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Figure6.1.1.Constructing a \(\{_{2}^{5}\}\)-star polygon.
The notation \(\{_{2}^{5}\}\) signifies that the star polygon has 5 points where every second point is connected. To create \(\{_{1}^{5}\}\text{,}\) we sketch a continuous path that connects each of the five points in Figure 6.1.1.(a) to the first point from it (its neighbor). The resulting figure does not look like a star but rather the convex regular pentagon shown in Figure 6.1.2.
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Figure6.1.2.The \(\{_{1}^{5}\}\)-star polygon.
Our study of star polygons is not limited to figures built with five or six concentric dots.
Definition6.1.3.
\(\{_{k}^{n}\}\) Star polygon on \(n\) points
We define the star polygon \(\{_{k}^{n}\}\) to be the planar figure formed by a set of \(n\) concentric points together with a series of line segments that join each point to the \(k^\text{th}\) point from it moving clockwise around the circle.
For a shorter phrasing we can call \(k\) the skip number of the star polygon on \(n\) points, but we need to take care. We are only skipping \(k-1\) points, connecting every \(k^\text{th}\) point.
Checkpoint6.1.4.
Sketch the \(\{_{3}^{5}\}\)- and \(\{_{4}^{5}\}\)-star polygons. What do you notice about them?
Hint.
Going to the third point in clockwise direction is the same as going to the second point in the counterclockwise direction.
Answer.
The \(\{_{3}^{5}\}\)-star polygon is identical to the \(\{_{2}^{5}\}\)-star polygon. The \(\{_{4}^{5}\}\)-star polygon, like the \(\{_{1}^{5}\}\)-star polygon, is a regular pentagon.
Similarly, we can draw star polygons on six vertices. The \(\{_{1}^{6}\}\)-star polygon is simply a regular hexagon.
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Figure6.1.5.The \(\{_{1}^{6}\}\)-star polygon.
As we create \(\{_{2}^{6}\}\) skipping every other vertex, we return to the starting point after creating a triangle. The three even-numbered vertices have not been visited.
We now repeat this process starting at an unvisited vertex and joining every second vertex until we return to this new starting vertex. As shown in Figure 6.1.7, this creates a second triangle congruent to the first but rotated 60 degrees about the center of the circle containing the six vertices. To aid in our exploration, a new color was used to draw the second triangle. You are also encouraged to use a new color each time you need to lift your pencil.
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Figure6.1.7.The \(\{_{2}^{6}\}\) star polygon.
Definition6.1.8.
Each figure created by connecting a (sub)set of points together by starting at one point and drawing a series of line segments together in one continuous process without lifting our pencil and turning only at the given points is a connected component 1
The term connected component comes from Graph Theory where points are called vertices and line segments are called edges.
. A connected component is a subset of points which are connected by line segments in the figure whose end points are those points. Two connected components will not have any endpoints in common. In Figure 6.1.7, there are two connected components both of which are triangles. One triangle connects the points 1, 3, and 5 and the other connects the points 2, 4, 6.
A connected component of a star polygon is also a star polygon. It may be the entire star polygon, a star polygon with fewer points, or a single line segment, namely the degenerate star polygon \(\{_{1}^{2}\}\text{.}\)
Checkpoint6.1.9.
(a)
What happens when you try to create a \(\{_{3}^{6}\}\)-star polygon by joining every third point? Is it a polygon, a star, overlapping polygons, or something else?
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Figure6.1.10.Template for drawing a six-vertex star polygon.
Answer.
The star polygon \(\{_{3}^{6}\}\) looks like three line segments intersecting at their midpoints.
(b)
Predict what \(\{_{4}^{6}\}\) and \(\{_{5}^{6}\}\) will look like. Explain.
Answer.
The star polygon \(\{_{4}^{6}\}\) will look like \(\{_{2}^{6}\}\text{,}\) namely two equilateral triangles rotated 60 degrees from each other. Also, \(\{_{5}^{6}\}\cong \{_{1}^{6}\}\) and is a regular hexagon.
Subsection6.1.2Exploring Star Polygons with Seven to Ten Vertices
We have observed that star polygons may look like regular polygons, overlapping rotated copies of regular polygons, or an intersection of line segments. One more possibility will be discovered later. After completing the exploration, you should be able to predict what a \(\{_{k}^{n}\}\) star polygon will look like based on the numbers \(n\) and \(k\text{.}\) Make conjectures and test them as you work through the exploration.
Exploration6.1.1.Star Polygons \(\{_{k}^{n}\}\) for \(7\lt n\lt 10\).
(a)Seven-dot star polygons.
Construct the star polygons \(\{_{1}^{7}\}\text{,}\)\(\{_{2}^{7}\}\text{,}\) and \(\{_{3}^{7}\}\text{.}\) Templates are provided in Figure 6.1.11.
If you are unable to sketch on the templates above, you may use the GeoGebra interactive in Figure 6.1.12 to draw 7-dot star polygons.
Figure6.1.12.GeoGebra interactive for drawing 7-dot star polygons
(b)
Match the star polygon on the left with the star polygon on the right that is the same.
Consider which five-dot and six-dot star polygons looked the same. Why? If you are not sure, construct the remaining seven-dot star polygons.
\(\{_{4}^{7}\}\)
\(\{_{3}^{7}\}\)
\(\{_{5}^{7}\}\)
\(\{_{2}^{7}\}\)
\(\{_{6}^{7}\}\)
\(\{_{1}^{7}\}\)
(c)Eight-dot star polygons..
(i)
Describe or sketch the star polygon \(\{_{1}^{8}\}\text{.}\)
(ii)
Sketch the star polygon \(\{_{2}^{8}\}\) using either the template in Figure 6.1.13 or the GeoGebra applet in Figure 6.1.14. Be sure to label your design with the symbol \(\{_{2}^{8}\}\text{.}\)
Draw star polygons by beginning each new segment at the endpoint of the previous segment. When you return to the starting point, switch to a new color. The connected components, formed by connecting points in this fashion, will be easy to identify.
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Figure6.1.13.Template for exploring 8-point star polygons.
Figure6.1.14.GeoGebra interactive for drawing 8-dot star polygons
What is the shape of the connected component containing point 1?
(v)
Summarize your responses in Task 6.1.1.c.iii and Task 6.1.1.c.iv to describe in words what a \(\{_{2}^{8}\}\) star polygon looks like.
Hint.
Your answer may include the number of copies of a familiar shape. You might also include a rotational angle or how these copies are related in position.
(vi)
Use copies of the template or the Geogebra applet to draw the remaining 8-dot star polygons. When two eight-dot star polygons are known to look identical, you do not need to draw the design a second time. Instead write the two symbols, \(\{_{k}^{8}\}\) below the design, replacing \(k\) by the appropriate number.
(vii)
Describe \(\{_{3}^{8}\}\text{.}\) Did you need to lift your pencil to draw it?
(viii)
Describe \(\{_{4}^{8}\}\text{.}\) Did you need to lift your pencil to draw it?
(d)Nine-dot Star Polygons.
Now that we have experience drawing star polygons, let us make some predictions for nine-dot star polygons.
(i)
Identify any nine-dot star polygons that are regular nonagons.
(ii)
Which nine-dot star polygons can be drawn without lifting one’s pencil?
(iii)
One of the nine-dot star polygons that requires us to lift our pencil and restart at a new dot is \(\{_{3}^{9}\}\text{.}\)
(A)
What shape is formed before we have to lift up our pencil?
(B)
How many copies of that shape appear in \(\{_{3}^{9}\}\text{?}\)
(C)
Will any of the 9-point star polygons look like line segments meeting at their midpoints? If so, identify the skip value(s), \(k\) for which this happens. If not, explain why this will not happen.
(iv)
Use the nine-dot template in Figure 6.1.15.(a) or the Geogebra applet in Figure 6.1.16 to check your answers to the above tasks and to answer the next question.
Are the star polygons \(\{_{2}^{9}\}\) and \(\{_{4}^{9}\}\) the same? Explain.
Figure6.1.16.GeoGebra interactive for drawing 9-dot star polygons
(e)Ten-dot Star Polygons.
Some of the ten-point star polygons are easy to predict but with ten points we will encounter one final possibility for the first time.
As with nine dots, we try to make predictions for some of the ten-point star polygons first. In your descriptions, be sure to identify the number of connected components and the shape of each connected component. A ten-dot template is provided following the predictions to explore the less obvious cases. Feel free to use the template for visualizing and verifying the anticipated star polygons as needed.
(i)
Describe \(\{_{1}^{10}\}\text{.}\) For what other value of \(k\text{,}\) is \(\{_{k}^{10}\}\cong\{_{1}^{10}\}\) ?
(ii)
Describe \(\{_{2}^{10}\}\text{.}\) This will have more than one connected component so be sure to identify the number of shape of the components. For what other value of \(k\text{,}\) is \(\{_{k}^{10}\}\cong\{_{2}^{10}\}\) ?
(iii)
Will any of the 10-point star polygons look like line segments meeting at their midpoints? If so, identify the skip value(s), \(k\) for which this happens. If not, explain why this will not happen.
(iv)
Either sketch or describe \(\{_{3}^{10}\}\text{.}\) You may use the template in Figure 6.1.17
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Figure6.1.17.Template for exploring 10-point star polygons.
Figure6.1.18.GeoGebra interactive for drawing 10-dot star polygons
(v)
In this task, we encounter a somewhat new result. We break our construction into separate steps to observe the features of the design.
(A)
Without lifting your pencil, connect every fourth dot until you return to your starting dot. What figure do you see? Describe this in terms of a previously constructed \(\{_{k}^{n}\}\) star polygon, identifying the values of \(n\) and \(k\text{.}\)
Hint.
Ignore the dots that are not part of this connected component. The value of \(n\) will be the number of dots in the connected component.
(B)
Now drawing the remaining connected component(s), each starting at a previously unused dot. If possible, use a different color for each connected component. How many connected components does \(\{_{4}^{10}\}\) have? Are the connected components congruent to each other?
Subsection6.1.3Making Conjectures and Generalizations about Star Polygons
By now, you may be able to predict what a star polygon \(\{_{k}^{n}\}\)looks like simply by analyzing relationships between \(n\) and \(k\text{.}\) In Exploration 6.1.2, we formalize these conjectures. When we make general conjectures and statements, we use variables instead of specific values.
Exploration6.1.2.
Throughout this exploration, the reader is invited to look back at their earlier work in this section. Templates for twelve-dot, fifteen-dot, twenty-dot, and twenty-four-dot circles are provided to allow students to further explore and validate their conjectures. Students wishing to use GeoGebra will find apps for 12-, 15-, 20-, and 24-dot circles at the end of the exploration. These can also be accessed via hyperlinks in the reading.
(a)
Let us begin by summarizing what we know about \(\{_{1}^{n}\}\text{.}\)
(i)
If \(n\) is any natural number greater than two, what is the shape of \(\{_{1}^{n}\}\text{?}\)
(ii)
Observing a pattern is not a mathematical proof. Justify your claim in Task 6.1.2.a.i by writing a few sentences that explain why this type of shape appears whenever \(k=1\) regardless of the choice of \(n\geq 3\text{.}\)
Hint.
How are the dots connected in \(\{_{1}^{n}\}\text{?}\)
(iii)
We observed that \(\{_{3}^{5}\}\cong \{_{2}^{5}\}\) and \(\{_{4}^{5}\}\cong\{_{1}^{5}\}\) in Checkpoint 6.1.4. Fill in the blank to complete the conjecture: \(\{_{j}^{n}\}\cong\{_{k}^{n}\}\) when \(j=\)
Hint.
Write your answer as an algebraic expression involving the variables \(k\) and \(n\text{.}\)
(iv)
Review the sketches of star polygons than have appeared in your reading and work. Which of the following consist of a single connected component connecting every dot before one needs to lift their pencil?
\(\{_{2}^{5}\}\)
\(\{_{1}^{6}\}\)
\(\{_{2}^{6}\}\)
\(\{_{3}^{6}\}\)
\(\{_{2}^{7}\}\)
\(\{_{3}^{7}\}\)
\(\{_{2}^{8}\}\)
\(\{_{3}^{8}\}\)
\(\{_{4}^{9}\}\)
\(\{_{6}^{9}\}\)
\(\{_{4}^{10}\}\)
\(\{_{7}^{10}\}\)
(v)
Which of the following star polygons do you expect to consist of a single connected component?
\(\{_{6}^{12}\}\)
\(\{_{7}^{20}\}\)
\(\{_{8}^{20}\}\)
(vi)
Based on Task 6.1.2.a.iv, make a conjecture as to the relationship between \(k\) and \(n\) when a star polygon consists of a single connected component. Be sure that all of the checked star polygons in Task 6.1.2.a.iv satisfy your relationship. Also, verify that any unchecked star polygons do not meet your criteria.
(vii)
Explain why drawing any star polygon which does not meet your criteria in Task 6.1.2.a.vi, for example \(\{_{8}^{20}\}\text{,}\) will require you to lift your pencil. Write your explanation so that it discusses general values of \(n\) and \(k\text{.}\) You may also include specific examples to make your argument clearer.
(viii)
Explain why drawing any star polygon which meets your criteria in Task 6.1.2.a.vi will allow you to connect every dot before you need to lift your pencil. Write your explanation so that it discusses general values of \(n\) and \(k\text{.}\) You may also include specific examples to make your argument clearer.
(ix)
Next we will focus on the shape of connected components of star polygons.
(A)
Sketch the star polygons \(\{_{4}^{12}\}\) and \(\{_{5}^{15}\}\) using the templates in Figure 6.1.19 or the GeoGebra apps in Figure 6.1.23 and Figure 6.1.24.
(a)Twelve-dot template.
(b)Fifteen-dot template.
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Figure6.1.19.Templates for 12-dot and 15-dot star polygons.
What shape is the initial connected component of \(\{_{4}^{12}\}\) and \(\{_{5}^{15}\}\text{?}\) Do the second sketched connected components have the same shape as the first?
(B)
Identify another pair of numbers \(n\) and \(k\) such that a connected component of \(\{_{k}^{n}\}\) has the shape you identified in Task 6.1.2.a.ix.A
(C)
Write a conjecture which identifies how the values \(n\) and \(k\) tell you that the first connected component has this shape. Include the name of this shape in your statement.
(D)
What is the shape of the first connected component of \(\{_{8}^{12}\}\text{?}\) Revise your conjecture in Task 6.1.2.a.ix.C to include this and similar cases.
(E)
Given the numbers \(n\) and \(k\text{,}\) can you determine how many copies of this shape will appear in the complete \(\{_{k}^{n}\}\) star polygon? How?
(F)
Next consider the connected components of \(\{^{12}_{3}\}\) and \(\{_{5}^{20}\}\text{.}\) What do you expect the shape of each connected component to be? Sketch these two star polygons, \(\{^{12}_{3}\}\) and \(\{_{5}^{20}\}\text{,}\) using different colors for each connected component. GeoGebra apps for these constructions may be accessed via Figure 6.1.23 and Figure 6.1.25.
(a)Twelve-dot template.
(b)Twenty-dot template.
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Figure6.1.20.Templates for 12-dot and 20-dot star polygons.
(G)
Did you correctly predict the shape of the connected component? What shape?
(H)
Identify another pair of numbers \(n\) and \(k\) for which the connected components of \(\{_{k}^{n}\}\) have this shape. You may identify one encountered already or one that you have not seen yet. How many copies of this shape will \(\{_{k}^{n}\}\) have?
(I)
Sketch the star polygons, \(\{_{6}^{15}\}\) and \(\{_{8}^{20}\}\text{.}\)
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Figure6.1.21.Twelve-dot template.
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Figure6.1.22.Twenty-dot template.
(A)
What shape are their individual connected components?
(B)
Denote the connected component’s shape using \(\{_{j}^{m}\}\) star polygon notation. What are the values of \(m\) and \(j\text{?}\)
(C)
How many connected components does \(\{_{6}^{15}\}\) have?
(D)
How many connected components does \(\{_{8}^{20}\}\) have?
(E)
How is the number of connected components related to the values \(n\) and \(k\text{?}\)
(x)
Summarize what you have learned about how the design of a \(\{_{k}^{n}\}\) star polygon is determined by the values of \(n\) and \(k\text{.}\) Include the number of connected components and the shape of the connected components.
Geogebra interactives are provided here for students wishing to use them.
Figure6.1.23.GeoGebra interactive for drawing 12-dot star polygonsFigure6.1.24.Geogebra interactive for drawing 15-point star polygonsFigure6.1.25.Geogebra interactive for drawing 20-point star polygons
(xi)
The following GeoGebra interactive can be used to verify that your claims hold for star polygons with 18 or 24 dots.
Figure6.1.26.Geogebra interactive for drawing 18- and 24-point star polygons
Exercises6.1.4Exercises
Building Our Toolbox
1.
In your own words, write instructions that a friend not in this class could use to draw a star polygon.
2.
Suppose \(a\) and \(b\) are any two positive integers.
(a)
Write a definition for \(GCF(a,b)\) the greatest common factor of \(a\) and \(b\text{.}\)
(b)
Use your definition in Task 6.1.4.2.a to explain why 12 is the greatest common factor of 84 and 120.
Hint.
You may use the words greatest and factor or synonyms in your response. Think about why 6 is not GCF(84,120). Why is GCF(84,120) not 24?
3.
Suppose that \(n\) is any natural number greater than 2. What is the shape of \(\{_{1}^{n}\}?\)
Hint.
Your answer will contain the variable \(n\text{.}\)
4.
Suppose \(\{_{j}^{n}\}\cong\{_{k}^{n}\}\text{.}\) What does that tell us about the relationship between \(j\text{,}\)\(k\text{,}\) and \(n?\)
\(n+k=j.\)
\(n=k\cdot j.\)
\(n=k+j.\)
\(k=n-j.\)
\(j=GCF(n,k).\)
5.
Suppose \(d\) is the greatest common factor of \(n\) and \(k.\)
(a)
How many connected components does \(\{_{k}^{n}\}\) have?
(b)
Each connected component of \(\{_{k}^{n}\}\) is a star polygon \(\{_{j}^{m}\}\text{.}\) What are the values of \(m\) and \(j?\)
Hint.
Your answers will be mathematical expressions involving \(n\) and \(k\text{.}\)
Skills and Recall
6.
Match each value of \(k\) to the corresponding description of \(\{_{k}^{42}\}.\)
\(k=4\)
Two copies of a 21-pointed star.
\(k=5\)
A 42-point star.
\(k=6\)
Six copies of a regular 7-gon.
\(k=7\)
Seven copies of a regular hexagon
\(k=9\)
Three copies of a 14-pointed star
7.
Identify a value of \(k\neq 14\) such that \(\{_{k}^{85}\}\cong \{_{23}^{85}\}\text{.}\)
8.
(a)
Determine all possible values of \(k\) such that \(\{_{k}^{55}\}\) is a regular polygon.
9.
(a)
Determine all possible values of \(k\lt 45\) such that \(\{_{k}^{90}\}\) consists of multiple copies of some regular polygon.
Hint.
There are eight values!
10.
Describe \(\{_{42}^{60}\}\) by identifying the number of connected components and their shape.
11.
A star polygon \(\{_{k}^{n}\}\) consists of five copies of \(\{_{3}^{16}\}\) sharing a common center.
(a)
Determine the value of \(n.\)
(b)
Determine all possible values of \(k\lt n\) such that \(\{_{k}^{n}\}\) has five copies of \(\{_{3}^{16}\}.\)
Extending the Concepts
12.
In this exercise, we will explore the relationship between star polygons and polygons.
(a)
A simple polygon is a polygon that does not intersect itself. In other words, the edges of a simple polygon will only intersect at their endpoints. Which, if any, star polygons are simple polygons?
(b)
Show that you can ‘simplify’ the star polygon \(\{_{2}^{5}\}\text{,}\) shown in Figure 6.1.1.(b), by placing an additional vertex at each point of intersection and erasing internal segments of edges. Remember that polygons may be convex 1.2.2 or concave 1.2.2.
(c)
What shape was formed by the segments you erased?
(d)
How many sides does the simplified polygon have?
13.
The star polygon \(\{_{3}^{8}\}\) is given in Figure 6.1.27.
(a)
What shape is in the center of the star polygon?
f(t)=(-sin(2*pi*(1+t)/8),cos(2*pi*(1+t)/8))
Figure6.1.27.The star polygon \(\{_{3}^{8}\}\text{.}\)
Writing Prompts
14.
Louie claims that the Greatest Common Factor of two numbers must be larger than their Least Common Multiple because of the words ‘greater’ and ‘less’. Huey disagrees and claims that the Least Common Multiple must be at least as large as their Greatest Common Factor.
Which is correct? Write a letter (perhaps to Louie or Huey) supporting your argument. Although you can use specific examples as partial support, you should include a more general argument.
