# Anomalous Mirror Symmetry Generated by Optical Illusion

## Abstract

**:**

## 1. Introduction

## 2. Mirror Symmetry and Anomalous Symmetry

## 3. Ambiguous Cylinders

## 4. Creation of Anomalous Mirror Symmetry

**Proposition.**

**Proof.**

## 5. Examples

## 6. Concluding Remarks

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 3.**Different views of the same cylinder as in Figure 2 rotated around a vertical axis. (

**a**) Object and its mirror image after 45 degrees rotation; (

**b**) Object and its mirror image after 90 degrees rotation; (

**c**) Object and its mirror image after 135 degrees rotation.

**Figure 4.**Swept surface and its interpretations: (

**a**) Surface swept by a vertical line segment; (

**b**) Plane-cut interpretation of the upper edge; (

**c**) Another interpretation which is mathematically possible but empirically unnatural.

**Figure 5.**Pair of plane curves that creates anomalous symmetry: (

**a**) Arbitrary closed curve compound of upper and lower curves both of which are monotone in the x direction; (

**b**) Closed curve that is point symmetric to the curve (a); (

**c**) Mirror image of (b) with respect to the mirror parallel to the $xz$ plane.

**Figure 8.**Fish-shape object creating anomalous mirror symmetry. (

**a**) Original posture; (

**b**) After 30 degrees rotation; (

**c**) After 90 degrees rotation; (

**d**) After 135 degrees rotation.

**Figure 10.**Car-shape object creating anomalous mirror symmetry. (

**a**) Original posture; (

**b**) After 30 degrees rotation; (

**c**) After 90 degrees rotation; (

**d**) After 135 degrees rotation.

**Figure 12.**Ship-shape object creating anomalous mirror symmetry. (

**a**) Original posture; (

**b**) After 30 degrees rotation; (

**c**) After 90 degrees rotation; (

**d**) After 135 degrees rotation.

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**MDPI and ACS Style**

Sugihara, K. Anomalous Mirror Symmetry Generated by Optical Illusion. *Symmetry* **2016**, *8*, 21.
https://doi.org/10.3390/sym8040021

**AMA Style**

Sugihara K. Anomalous Mirror Symmetry Generated by Optical Illusion. *Symmetry*. 2016; 8(4):21.
https://doi.org/10.3390/sym8040021

**Chicago/Turabian Style**

Sugihara, Kokichi. 2016. "Anomalous Mirror Symmetry Generated by Optical Illusion" *Symmetry* 8, no. 4: 21.
https://doi.org/10.3390/sym8040021