Lesson 14: Perspective and Symmetry

class: center, middle, inverse, title-slide

.title[
# Lesson 14: Perspective and Symmetry
]
.author[
### Fei Ye
]
.date[
### May, 2024
]

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## Unit 11B: Perspective and Symmetry

*Primary Source:* PPT for the book "Using & Understanding Mathematics".

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## Connection Between Visual Arts and Mathematics

At least three aspects of the visual arts relate directly to mathematics:

- Perspective

- Symmetry

- Proportion

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## Side View of a Hallway

.center[
![](data:image/png;base64,#img/Perspective.png)

Side view of a hallway, showing perspectives.
]

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## Artist's View of the Hallway

.center[
![](data:image/png;base64,#img/VanishingPoint.png)
]

An artist's view of the hallway, showing perspectives. The lines `$L_1$`, `$L_2$`, `$L_3$`, and `$L_4$`, which are parallel in the actual scene, are not parallel in the painting.

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## Perspective

- Lines that are parallel in the actual scene, but not parallel in the painting, meet at a single point, P , called the **principle vanishing point**.

- All lines that are parallel in the real scene and perpendicular to the canvas must intersect at the principal vanishing point of the painting.

- Lines that are parallel in the actual scene but not perpendicular to the canvas intersect at their own vanishing point, called the **horizon line**.

---

## Example: Perspective

Leonardo da Vinci (1452-1519) contributed greatly to the science of perspective. We can see da Vinci's mastery of perspective in many of his paintings. If you study *The Last Supper*, you will notice several parallel lines in the actual scene intersecting at the principal vanishing point of the painting, which is directly behind the central figure of Christ.
.center[
![:resize , 60%](data:image/png;base64,#img/TheLastSupper.png)
]

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## Example: Woodcut (1/2)

The German artist Albrecht Dürer (1471-1528) further developed the science of perspective.

The next slide shows one of Dürer's woodcuts, showing an artist using his principles of perspective.

A string from a point on the lute is attached to the wall at the point corresponding to the artist's eye. At the point where the string passes through the frame, a point is placed on the canvas. As the string is moved to different points on the lute, a drawing of the lute is created in perfect perspective on the canvas.

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## Example: Woodcut (2/2)

.center[
![](data:image/png;base64,#img/Woodcut.png)
]

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## Symmetry

**Symmetry** refers to a kind of balance, or a repetition of patterns.

In mathematics, symmetry is a property of an object that remains unchanged under certain operations.

.row[
.onethird-left[
**Reflection symmetry**: An object remains unchanged when reflected across a straight line.

![](data:image/png;base64,#img/Reflection.png)
]
.onethird-center[
**Translation symmetry**: A pattern remains the same when shifted to the left or to the right.

![](data:image/png;base64,#img/Translation.png)
]
.onethird-right[
**Rotation symmetry**: An object remains unchanged when rotated through some angle about a point.

![](data:image/png;base64,#img/Rotation.png)
]
]

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## Angel and Order of Rotational Symmetry

The **angle of rotational symmetry** of an object is the smallest angle of rotation that preserves the object.

The **order of rotational symmetry** of an object is that the number of times that the rotated object matches the original object during a full rotation of 360 degrees.

Angle and order of rational symmetry satisfy the following equality.

`$$\text{angle of rotaion}\times \text{order of raotaion}=360\text{ deg}$$`
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## Visualization of Rotational Symmetry

.pull-left[
.center[
			<iframe src="https://www.mathsisfun.com/geometry/isym-rot2.html" scrolling="no" style="width:250px; height:250px; overflow:hidden; margin:auto;  border: none;"></iframe><br>
<span>Rotational Symmetry<br>
Order 2</span>
]
]
.pull-right[
.center[
<iframe src="https://www.mathsisfun.com/geometry/isym-rot3.html" scrolling="no" style="width:250px; height:250px; overflow:hidden; margin:auto;  border: none;"></iframe><br>
<span>Rotational Symmetry<br>
Order 3</span>
]
]

.footmark[
Source: [Mathisfun](https://www.mathsisfun.com/geometry/symmetry-rotational.html)
]

---

## Example: Finding Symmetries

.pull-left[
Identify the types of symmetry in the star.
]
.pull-right[
![:resize , 25%](data:image/png;base64,#img/Star.png)
]

**Solution:** The five-pointed star has five lines about which it can be flipped (reflected) without changing its appearance, so it has five reflection symmetries.

Because it has five vertices that all look the same, it can be rotated by 1/5 of a full circle, or `$360^\circ/5 = 72^\circ$`, and it still looks the same. Similarly, its appearance remains unchanged if it is rotated by `$2 \times 72^\circ= 144^\circ$`, `$3 \times 72^\circ = 216^\circ$`, `$4 \times 72^\circ = 288^\circ$` or `$5 \times 72^\circ = 360^\circ$`

Angle and order of the rotational symmetry of the star are `$72^\circ$` and 5 respectively.

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## Tilings (Tessellations) (1/2)

A tiling is an arrangement of polygons that interlock perfectly without overlapping.

.center[
Regular Polygon Tessellations

![](data:image/png;base64,#img/Tilings.png)
]

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## Tilings (Tessellations) (2/2)

- Some tilings use irregular polygons.

- Tilings that are periodic have a pattern that is repeated throughout the tiling.

- Tilings that are aperiodic do not have a pattern that is repeated throughout the entire tiling.

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## Example: Quadrilateral Tiling (1 of 2)

Create a tiling by translating the quadrilateral shown. As you translate the quadrilateral, make sure that the gaps left behind have the same quadrilateral shape.

.center[
![](data:image/png;base64,#img/quadrilateral.png)
]

**Solution:** We can find the solution by trial and error, translating the quadrilateral in different directions until we have correctly shaped gaps. The figure on the next slide shows the solution.

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## Example: Quadrilateral Tiling (2 of 2)

Note that the translations are along the directions of the two diagonals of the quadrilateral. The gaps between the translated quadrilaterals are themselves quadrilaterals that interlock perfectly to complete the tiling.
.center[
![](data:image/png;base64,#img/QuadrilateralTiling.png)
]

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## Practice: Drawing MATH with Perspective

Make the letters M, A, T, and H into three-dimensional solid letter.

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## Practice: Symmetry in Letters

Find all of the capital letters of the alphabet that have a rotation symmetry.

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## Practice: Rotational Symmetry of Quadrilaterals

Determine the order and angle of rotational symmetry of each quadrilateral.

.footmark[
Source: [https://www.geogebra.org/m/pmJ656a6](https://www.geogebra.org/m/pmJ656a6)
]

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## Practice: Tiling from Translating Triangles

Make a tiling from an isosceles triangle.