Question: What are the characteristics of an image formed by a plane mirror?
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In general, the characteristics of a plane mirror image can be described as follows:
1. The image is left-right reversed or front-back reversed in relation to the object.
2. The image is virtual because it cannot be formed on a screen.
3. The image has the same size as the object.
4. The image is upright.
5. The image is located at the same distance behind the mirror as the object is in front of the mirror. (Image distance = object distance)
6. The image has the same color as the object.
It has been controversial whether the nature of image formed by a plane mirror should be described as “left-right reversed” or “front-back reversed.” This is related to an interesting question, why does a mirror reverse left and right but not up and down?Currently, the plane mirror image could be specified in textbooks as “lateral inverted” (Muncaster, 1993), “left-right reversed” (Cutnell & Johnson, 2004), “appears left-right reversed” (Giancoli, 2005), “front-back reversed” (Knight, 2004), or “depth inverted” (Tipler & Mosca, 2004). Interestingly, Tomonaga, who shared the 1965 Nobel (Physics) Prize with Feynman and Schwinger, discussed the mirror reflection problem with his colleagues and believed that the top–bottom and front–back axes had absoluteness in a “psychological space” (Tabata & Okuda, 2000; Tomonaga, 1965). However, there is no agreement on the description and explanation of the plane mirror image.
The concept of the plane mirror image is related to terms such as “parity,” “enantiomorph,” and “chirality.” For example, Lord Kelvin defines the concept of chirality in a footnote of a lecture, titled The molecular tactics of a crystal. This famous footnote reads:
“I call any geometrical figure, or group of points, chiral, and say that it has chirality, if its image in a plane mirror, ideally realized, cannot be brought to coincide with itself. Two equal and similar right hands are homochirally similar. Equal and similar right and left hands are heterochirally similar or ‘allochirally’ similar (but heterochirally is better). These are also called ‘enantiomorphs,’ after a usage introduced, I believe, by German writers. Any chiral object and its image in a plane mirror are heterochirally similar (Kelvin, 1894, p. 27).”
How would Feynman answer?
Feynman’s answer may include the concepts “front-back reversed” and “handedness of an object” pertaining to the nature of plane mirror image. However, it is more meaningful to understand his explanations on “front-back reversed,” “handedness of an object,” and “definitions of left and right.”
Feynman’s answer may include the concepts “front-back reversed” and “handedness of an object” pertaining to the nature of plane mirror image. However, it is more meaningful to understand his explanations on “front-back reversed,” “handedness of an object,” and “definitions of left and right.”
1. Front-back reversed: During a BBC Television interview, Feynman (1994) explains that “if you wave one hand, then the hand in the mirror that waves is opposite it – the hand on the ‘east’ is the hand on the ‘east,’ and the hand on the ‘west’ is the hand on the ‘west.’ The head that’s up is up, and the feet that are down are down. So everything’s really all right. But what’s wrong is that if this is ‘north,’ then your nose is to the back of your head, but in the image, the nose is to the ‘south’ of the back of your head. What happen is, the image has neither the right nor the left mixed up with the top and the bottom, but the front and the back have been reversed, you see (p. 37).” In a sense, there is a semantic problem in describing the nature of plane mirror image (Ansbacher, 1992). It is remarkable that Feynman is not constrained by the words, “left” and “right,” and he is able to replace them by “east” and “west.” Thus, the plane mirror image can be simply described as either “north-south reversed” or “front-back reversed.”
In general, the choice of phrase such as “front-back reversed” is imprecise. To be more precise, “[a] mirror image reproduces exactly all object points in two spatial directions parallel to the mirror surface, but reverses the sequential ordering of object points in the direction of the third spatial axis, perpendicular to the mirror plane (Galili & Goldberg, 1993, p. 463).” In other words, a plane mirror does not vary the coordinates such as y and z in the two-dimensional planes that are parallel to the mirror, but it reverses the “x” coordinates, for example, that are in the same direction as the axis of the mirror. However, physics teachers may find it cumbersome to describe the nature of plane mirror image in greater detail.
In short, the nature of plane mirror image may appear as “left-right reversed,” “top-bottom reversed,” and “front-back reversed” (See Fig 1a, Fig 1b, and Fig 1c). According to Feynman (1994), “we say left and right are interchanged, but really the symmetrical way is it’s along the axis of the mirror that things get interchanged (p. 38).” The descriptions of the plane mirror image are dependent on the “axis of the mirror.” In Fig 1a, the plane mirror image of a right-handed glove appears as “left-right reversed.” The axis of this mirror is in a horizontal direction and the mirror is placed beside the glove. In Fig 1b, the plane mirror image of the upright glove appears as “top-bottom reversed.” The axis of the mirror is in a vertical direction and the mirror is placed below the glove. In Fig 1c, the plane mirror image of the glove appears as “front-back reversed.” The axis of this mirror is horizontal and the mirror is placed in front of the glove. Essentially, the location of the mirror relative to the object affects descriptions of the plane mirror image.
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Fig 1a. Mirror beside the object (“left-right reversed”)
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Fig 1b. Mirror below the object (“top-bottom reversed”)
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Fig 1c. Mirror in front of the object (“front-back reversed”)
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2. The handedness of an object: Although Feynman explains that a characteristic of the plane mirror image is front-back reversed, he also describes the handedness of an object. Feynman provides the following example: “[t]he first molecule, the one that comes from the living thing, is called L-alanine. The other one, which is the same chemically, in that it has the same kinds of atoms and the same connections of the atoms, is a ‘right-hand’ molecule, compared with the ‘left-hand’ L-alanine, and it is called D-alanine… (Left-handed sugar tastes sweet, but not the same as right-handed sugar.) So it looks as though the phenomena of life permit a distinction between ‘right’ and ‘left,’ or chemistry permits a distinction because the two molecules are chemically different (Feynman et al., 1963, section 52–4 Mirror reflections).” Simply phrased, the chemical and physical properties of the object are dependent on its handedness.
The handedness of a particle is an important concept which helps to understand the principle of the conservation of parity (mirror symmetry). As an analogy, Swedish physicist Cecilia Jarlskog identified a similarity between left-handed neutrinos and “vampire”: they do not have a mirror image (t’Hooft, 1997). In other words, the plane mirror changes the handedness of a particle, and this “mirror particle” may or may not be observed in nature. In essence, nature has a preference on the handedness of the particle, and it does not conform to the conservation of parity principle. Importantly, T. D. Lee and C. N. Yang (1956) predicted three experiments that illustrate the non-conservation of parity in weak interactions. It resolves the famous “tau-theta puzzle” pertaining to the decay of kaons, which supposedly have the same mass but they can decay into products of opposite parity.
The non-conservation of parity should not be simply illustrated by the handedness of a particle. It also involves physical conditions such as very low temperature and strong magnetic field. To quote Feynman, “When we put cobalt atoms in an extremely strong magnetic field, more disintegration electrons go down than up. Therefore, if we were to put it in a corresponding experiment in a “mirror,” in which the cobalt atoms would be lined up in the opposite direction, they would spit their electrons up, not down; the action is unsymmetrical (Feynman et al., 1963, section 52-7 Parity is not conserved!).” In this experiment, the observation of a preferred direction of decays helps to establish the violation of parity. It is pertinent to understand how the “mirror condition” such as the magnetic field is related to the handedness of the object (e.g. cobalt atoms).
3. Definitions of “left” and “right”: Feynman would discuss the problems of defining “left” and “right” and explain that “the world does not have to be symmetrical. For example, using what we may call ‘geography,’ surely ‘right’ can be defined. For instance, we stand in New Orleans and look at Chicago, and Florida is to our right… (Feynman et al., 1963, section 52–4 Mirror reflections).” That is, it is possible to define “right” and “left” by using geography because there is no symmetry between two locations such as Chicago and Florida. However, the directions for up-down, left-right and front-back are arbitrarily defined depending on one’s orientation and perspective. Generally speaking, definitions of left and right are ambiguous due to possible rotations of an observer about a vertical axis. Thus, one should initially define the directions of “up,” “down,” “front,” and “back.”
Interestingly, Feynman would explore how to tell a Martian the definitions of “left” and “right.” During a lecture at Cornell University, he mentions the following procedure: “take a radioactive stuff, a neutron, and look at the electron which comes from such a beta-decay. If the electron is going up as it comes out, the direction of its spin is into the body from the back on the left side. That defines left. That is where the heart goes (Feynman, 1965, p. 103).” Alternatively, in The Feynman Lectures on Physics, he describes the Wu et al. (1957) experiment: “build yourself a magnet, and put the coils in, and put the current on, and then take some cobalt and lower the temperature. Arrange the experiment so the electrons go from the foot to the head, then the direction in which the current goes through the coils is the direction that goes in on what we call the right and comes out on the left (Feynman et al., 1963, section 52-7 Parity is not conserved!).” Nevertheless, Feynman also defines the direction of “top” and “bottom” in this experiment.
Ideally, the definitions of “right” and “left” should not be dependent on history and convention (Feynman et al., 1963, section 52-4 Mirror reflections). As an example, most screws have right-handed threads which are arbitrarily determined. In fact, it is possible to have left-handed screws which are traditionally used for coffins (McManus, 2002). On the other hand, the right-handed rule for magnetic fields and definition of neutrinos as left-handed are merely conventions. Physicists could also define electric field as a pseudo-vector and magnetic field to be a vector (Griffiths, 2004). Similarly, physicists could have renamed neutrinos as anti-neutrinos and vice versa, thus changing their handedness. However, an interesting question now is whether right-handed neutrinos can be detected, and thus, they exist not only in the mirror world but also the real world.
In summary, we may describe the nature of a plane mirror image as “front-back reversed,” “left-right reversed,” and “top-bottom reversed.” The descriptions of the plane mirror image are dependent on the “handedness of an object,” and the “axis of the mirror.” However, we should understand Feynman’s reasonings pertaining to the concept of “front-back reversed,” “handedness of an object,” and “definitions of left and right.”
The handedness of a particle is an important concept which helps to understand the principle of the conservation of parity (mirror symmetry). As an analogy, Swedish physicist Cecilia Jarlskog identified a similarity between left-handed neutrinos and “vampire”: they do not have a mirror image (t’Hooft, 1997). In other words, the plane mirror changes the handedness of a particle, and this “mirror particle” may or may not be observed in nature. In essence, nature has a preference on the handedness of the particle, and it does not conform to the conservation of parity principle. Importantly, T. D. Lee and C. N. Yang (1956) predicted three experiments that illustrate the non-conservation of parity in weak interactions. It resolves the famous “tau-theta puzzle” pertaining to the decay of kaons, which supposedly have the same mass but they can decay into products of opposite parity.
The non-conservation of parity should not be simply illustrated by the handedness of a particle. It also involves physical conditions such as very low temperature and strong magnetic field. To quote Feynman, “When we put cobalt atoms in an extremely strong magnetic field, more disintegration electrons go down than up. Therefore, if we were to put it in a corresponding experiment in a “mirror,” in which the cobalt atoms would be lined up in the opposite direction, they would spit their electrons up, not down; the action is unsymmetrical (Feynman et al., 1963, section 52-7 Parity is not conserved!).” In this experiment, the observation of a preferred direction of decays helps to establish the violation of parity. It is pertinent to understand how the “mirror condition” such as the magnetic field is related to the handedness of the object (e.g. cobalt atoms).
3. Definitions of “left” and “right”: Feynman would discuss the problems of defining “left” and “right” and explain that “the world does not have to be symmetrical. For example, using what we may call ‘geography,’ surely ‘right’ can be defined. For instance, we stand in New Orleans and look at Chicago, and Florida is to our right… (Feynman et al., 1963, section 52–4 Mirror reflections).” That is, it is possible to define “right” and “left” by using geography because there is no symmetry between two locations such as Chicago and Florida. However, the directions for up-down, left-right and front-back are arbitrarily defined depending on one’s orientation and perspective. Generally speaking, definitions of left and right are ambiguous due to possible rotations of an observer about a vertical axis. Thus, one should initially define the directions of “up,” “down,” “front,” and “back.”
Interestingly, Feynman would explore how to tell a Martian the definitions of “left” and “right.” During a lecture at Cornell University, he mentions the following procedure: “take a radioactive stuff, a neutron, and look at the electron which comes from such a beta-decay. If the electron is going up as it comes out, the direction of its spin is into the body from the back on the left side. That defines left. That is where the heart goes (Feynman, 1965, p. 103).” Alternatively, in The Feynman Lectures on Physics, he describes the Wu et al. (1957) experiment: “build yourself a magnet, and put the coils in, and put the current on, and then take some cobalt and lower the temperature. Arrange the experiment so the electrons go from the foot to the head, then the direction in which the current goes through the coils is the direction that goes in on what we call the right and comes out on the left (Feynman et al., 1963, section 52-7 Parity is not conserved!).” Nevertheless, Feynman also defines the direction of “top” and “bottom” in this experiment.
Ideally, the definitions of “right” and “left” should not be dependent on history and convention (Feynman et al., 1963, section 52-4 Mirror reflections). As an example, most screws have right-handed threads which are arbitrarily determined. In fact, it is possible to have left-handed screws which are traditionally used for coffins (McManus, 2002). On the other hand, the right-handed rule for magnetic fields and definition of neutrinos as left-handed are merely conventions. Physicists could also define electric field as a pseudo-vector and magnetic field to be a vector (Griffiths, 2004). Similarly, physicists could have renamed neutrinos as anti-neutrinos and vice versa, thus changing their handedness. However, an interesting question now is whether right-handed neutrinos can be detected, and thus, they exist not only in the mirror world but also the real world.
In summary, we may describe the nature of a plane mirror image as “front-back reversed,” “left-right reversed,” and “top-bottom reversed.” The descriptions of the plane mirror image are dependent on the “handedness of an object,” and the “axis of the mirror.” However, we should understand Feynman’s reasonings pertaining to the concept of “front-back reversed,” “handedness of an object,” and “definitions of left and right.”
Note:
In an article titled Theory of the Fermi interaction, Feynman and Gell-Mann (1958) state that “only neutrinos with left-hand spin can exist (p. 195).”
In an article titled Theory of the Fermi interaction, Feynman and Gell-Mann (1958) state that “only neutrinos with left-hand spin can exist (p. 195).”
References:
1. Ansbacher, T. H. (1992). Left-Right Semantics. The Physics Teacher, 30(2), 70.
2. Cutnell J. D., & Johnson, K. W. (2004). Physics (6th ed.). New Jersey: John Wiley & Sons.
3. Feynman, R. P. (1965). The character of physical law. Cambridge: MIT Press.
3. Feynman, R. P. (1965). The character of physical law. Cambridge: MIT Press.
4. Feynman, R. P. (1994). No Ordinary Genius - The Illustrated Richard Feynman. New York: W. W. Norton and Company.
5. Feynman, R. P., & Gell-Mann, M. (1958). Theory of the Fermi interaction. Physical Review, 109(1), 193-198.
6. Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.
6. Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.
7. Galili I., & Goldberg F. (1993). Left-right Conversions in a Plane Mirror. The Physics Teacher, 31(8), 463–466.
8. Giancoli, D. C. (2005). Physics: Principles with Applications (6th ed.). Upper Saddle River, N. J.: Prentice Hall.
9. Griffiths, D. (2004). Introduction to Elementary Particles. Weinheim: Wiley-VCH.
10. Kelvin, W. T. (1894). The molecular tactics of a crystal. Oxford: Clarendon Press.
11. Knight, R. D. (2004). Physics for Scientists and Engineers with Modern Physics: A Strategic Approach. Boston: Addison Wesley.
12. Lee, T. D. & Yang, C. N. (1956). Question of parity conservation in weak interactions. Physical review, 104(1), 254–258.
13. McManus, C. (2002). Right Hand, Left Hand: The Origins of Asymmetry in Brains, Bodies, Atoms and Cultures. Cambridge: Harvard University Press.
14. Muncaster, R. (1993). A Level Physics (4th ed). Cheltenham: Nelson Thornes.
15. Tabata, T., & Okuda, S. (2000). Mirror reversal simply explained without recourse to psychological processes. Psychonomic Bulletin & Review, 7(1), 170–173.
16. t’Hooft, G. (1997). In Search of the Ultimate Building Blocks. Cambridge: Cambridge University Press.
17. Tipler, P. A., & Mosca, G. P. (2004). Physics for Scientists and Engineers (5th ed.). New York: W. H. Freeman.
18. Tomonaga, S. (1965). Kagaminonaka no sekai [The world in the mirror]. Tokyo: Misuzu-Shobo.
19. Wu, C. S., Ambler, E., Hayward, R. W., Hoppes, D. D., & Hudson, R. P. (1957). Experimental test of parity conservation in beta decay. Physical review, 105(4), 1413-1415.
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