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Chapter 9: Problem 16

Determine whether each regular polygon tessellates the plane. Explain. 23 -gon

Short Answer

Expert verified
A regular 23-gon does not tessellate the plane, as its interior angle doesn't divide evenly into 360 degrees.

Step by step solution

01

Understand Polygon Tessellation

A regular polygon tessellates the plane if copies of the polygon can cover a plane without any gaps or overlaps. Only specific regular polygons meet this criterion.
02

Consider Tessellation Angles

For a regular polygon to tessellate, the interior angle of the polygon must be a divisor of 360 degrees. This condition ensures that the angles can make a full circle around a point when the polygons are placed together.
03

Calculate Interior Angle

The interior angle of a regular polygon with \( n \) sides is given by the formula: \( \theta = \frac{(n-2) \times 180}{n} \). For a 23-gon, substitute \( n=23 \): \[ \theta = \frac{(23-2) \times 180}{23} = \frac{21 \times 180}{23} \approx 164.35 \text{ degrees} \]
04

Verify Divisibility by 360

Check if \( 164.35 \) is a divisor of 360 degrees: Since \( 360 \div 164.35 \approx 2.19 \), which is not an integer, 164.35 degrees does not divide 360 evenly.Therefore, the 23-gon cannot tessellate the plane.
05

Conclusion: Determine Tessellation Ability

A regular 23-gon does not tessellate the plane because its interior angle is not a divisor of 360 degrees, meaning it cannot form full circles around a point without leaving gaps or overlaps.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Regular Polygons
A regular polygon is a geometric shape with all sides and angles equal. Each of these polygons have unique characteristics that affect their ability to tessellate, or cover a plane without gaps or overlaps.
Common examples of regular polygons include triangles, squares, and hexagons, which famously tessellate well due to their specific angle properties.
When considering a less common polygon, like a regular 23-gon, the same ideas apply, but the feasibility of tessellation becomes complicated as the number of sides increases.
Interior Angles
The interior angle of a regular polygon is crucial in determining its ability to tessellate. The formula to calculate an interior angle \( \theta \) of a polygon with \( n \) sides is:
\( \theta = \frac{(n-2) \times 180}{n} \).
This angle must divide evenly into 360 degrees to form a tessellation. In our case with a 23-gon, this condition fails.
Calculating, we find that a 23-gon has an interior angle of approximately 164.35 degrees. Since 164.35 is not a divisor of 360, it can't fit together neatly without leaving gaps.
Geometry Education
Understanding tessellation in geometry education helps students visualize how shapes fit together in two dimensions.
This concept not only builds spatial awareness but also connects to other mathematical ideas like symmetry and pattern creation.
Such exercises involving more complex polygons challenge students to apply formulas and consider properties of numbers, enhancing both their calculation skills and understanding of geometric principles.

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