Parry Moon

Parry H. Moon
Parry H. Moon
Born	February 14, 1898; Beaver Dam, Wisconsin
Died	March 4, 1988 (aged 90)
Nationality	United States
Alma mater	University of Wisconsin; MIT
Known for	Contributions to electromagnetic field theory; Holors
Awards	1974 Illuminating Engineering Society's Gold Medal
	Scientific career
Fields	Electrical engineer
Institutions	MIT

Parry Hiram Moon (/muːn/; 1898–1988) was an American electrical engineer, who with Domina Eberle Spencer co-authored eight scientific books and over 200 papers on subjects including electromagnetic field theory, color harmony, nutrition, aesthetic measure, and advanced mathematics. He also developed a theory of holors.^[1]

Biography

Parry Hiram Moon was born in Beaver Dam, Wisconsin to Ossian C. and Eleanor F. (Parry) Moon. He received a BSEE from University of Wisconsin in 1922 and an MSEE from MIT in 1924. Unfulfilled with his work in transformer design at Westinghouse, Moon obtained a position as research assistant at MIT under Vannevar Bush. He was hospitalized for six months after sustaining injuries from experimental work in the laboratory. He later continued his teaching and research as an associate professor in MIT's Electrical Engineering Department. He married Harriet Tiffany, with whom he had a son. In 1961, after the death of his first wife, he married his co-author, collaborator and former student, Domina Eberle Spencer, a professor of mathematics. They have one son. Moon retired from full-time teaching in the 1960s, but continued his research until his death in 1988.

Scientific contributions

Moon’s early career focused in optics applications for engineers. Collaborating with Domina Eberle Spencer, he began researching electromagnetism and Amperian forces. The quantity of papers that followed culminated in Foundations of Electrodynamics,^[2] unique for its physical insights, and two field theory books, which became standard references for many years. Much later, Moon and Spencer unified the approach to collections of data (vectors, tensors, etc.), with a concept they coined as “holors”.^[1] Through their work, they became disillusioned with Einsteinian relativity and sought neo-classical explanations for various phenomena.

Holors

Moon and Spencer invented the term holor^[1]^[3]^[4] (/ˈhoʊlər/; Greek ὅλος "whole") for a mathematical entity that is made up of one or more independent quantities, or merates^[5] as they are called in the theory of holors. For example, holors include:

real numbers, complex numbers, quaternions, and other hypercomplex numbers;
scalars, vectors, and matrices;
(geometric) scalars, (geometric) vectors, and tensors;
non-tensorial geometric arrays of quantities such as the Levi-Civita symbol; and
non-tensorial non-geometric arrays of quantities such as neural network (node and/or link) values or indexed inventory tables.

To explain the usefulness of coining this term, Moon and Spencer wrote the following:

Holors could be called "hypernumbers," except that we wish to include the special case of $N=0$ (the scalar), which is certainly not a hypernumber. On the other hand, holors are often called "tensors." But this is incorrect, in general, for the definition of a tensor includes a specific dependence on coordinate transformation. To achieve sufficient generality, therefore, it seems best to coin a new word such as holor.
— Theory of Holors: A Generalization of Tensors^[1] (page 11)

Although the terminology relating to holors is not currently commonly found online, academic and technical books and papers that use this terminology can be found in literature searches (for instance, using Google Scholar). For example, books and papers on general dynamical systems,^[6] Fourier transforms in audio signal processing,^[7] and topology in computer graphics^[8] contain this terminology.

At a high level of abstraction, a holor can be considered as a whole -- as a quantitative object without regard to whether it can be broken into parts or not. In some cases, it may be manipulated algebraically or transformed symbolically without needing to know about its inner components. At a lower level of abstraction, one can see or investigate how many independent parts the holor can be separated into, or if it can't be broken into pieces at all. The meaning of "independent" and "separable" may depend upon the context. Although the examples of holors given by Moon and Spencer are all discrete finite sets of merates (with additional mathematical structure), holors could conceivably include infinite sets, whether countable or not (again, with additional mathematical structure that provides meaning for "made up of" and "independent"). At this lower level of abstraction, a particular context for how the parts can be identified and labeled will yield a particular structure for the relationships of merates within and across holors, and different ways that the merates can be organized for display or storage (for example, in a computer data structure and memory system). Different kinds of holors can then be framed as different kinds of general data types or data structures.

Holors include arbitrary arrays. A holor can be represented as an array of quantities, possibly a single-element array or a multi-element array with one or more indices to label each element. The context of the usage of the holor will determine what sorts of labels are appropriate, how many indices there should be, and what values the indices will range over. The representing array could be jagged (with different dimensionality per index) or of uniform dimensionality across indices. (An array with two or more indices is often called a "multidimensional array", referring to the dimensionality of the shape of the array rather than other degrees of freedom in the array. The term "multi-indexed" may be a less-ambiguous description. A multi-dimensional array is a holor, whether that refers to a single-indexed array of dimension two or greater, or a multi-element array with two or more indices.) A holor can thus be represented with a symbol and zero or more indices, such as $H^{ij}$ -- the symbol $H$ with the two indices $i$ and $j$ shown in superscript.

In the theory of holors, the number of indices $N$ used to label the merates is called the valence.^[9] This term is to remind one of the concept of chemical valence, indicating the "combining power" of a holor. (This "combining power" sense of valence is really only relevant in contexts where the holors can be combined, such as the case of tensor multiplication where indices pair up or "bond" to be summed-over.) For each index $i$ , there is a range of values the index may range over $n_{i}$ , which is called the plethos^[10] of that index, indicating the "dimensionality" related to that index. For a holor with uniform dimensionality over all of its indices, the holor itself can be said to have a plethos equal to the plethos of each index. (This term thus helps to resolve some of the ambiguity of referring to the "dimension" of a holor.) So, in particular, holors that are represented as arrays of hypercubic (or N-cubic) shape may be classified with respect to their plethos $n$ and valence $N$ .

If proper index conventions are maintained then certain relations of holor algebra are consistent with that of real algebra, i.e., addition and uncontracted multiplication are both commutative and associative. Holors can be either nongeometric objects or geometric objects. They further classify the geometric objects as either oudors or akinetors, where the (contravariant) akinetors^[11] transform as:

v^{i'}=\sigma {{\partial x^{i'}} \over {\partial x^{i}}}v^{i},

and the oudors^[12] contain all other geometric objects (such as Christoffel symbols). The tensor is a special case of the akinetor where $\sigma =1$ . Akinetors correspond to pseudotensors in standard nomenclature.

Moon and Spencer also provide a novel classification of geometric figures in affine space with homogeneous coordinates. For example, a directed line segment that is free to slide along a given line is called a fixed rhabdor^[13] and corresponds to a sliding vector^[14] in standard nomenclature. Other objects in their classification scheme include free rhabdors, kineors,^[15] fixed strophors,^[16] free strophors, and helissors.^[17]

Bibliography

Books

Parry Moon, The Scientific Basis of Illuminating Engineering, McGraw-Hill, 608pp. (1936) (ASIN B000J2QFAI).
Parry Moon, Lighting Design, Addison-Wesley Press, 191pp. (1948) (ASIN B0007DZUFA).
Parry Moon, A Proposed Musical Notation, (1952) (ASIN B0007JY81G).
Parry Moon & Domina Eberle Spencer, Foundations of Electrodynamics, D. Van Nostrand Co., 314pp. (1960) (ASIN B000OET7UQ).^[2]
Parry Moon & Domina Eberle Spencer, Field Theory for Engineers, D. Van Nostrand Co., 540pp. (1961) (ISBN 978-0442054892).
Parry Moon & Domina Eberle Spencer, Field Theory Handbook: Including Coordinate Systems, Differential Equations and Their Solutions, Spring Verlag, 236pp. (1961) (ISBN 978-0387184302).
Parry Moon & Domina Eberle Spencer, Vectors, D. Van Nostrand Co., 334pp. (1965) (ASIN B000OCMWTW).
Parry Moon & Domina Eberle Spencer, Partial Differential Equations, D. C. Heath, 322pp. (1969) (ASIN B0006DXDVE).
Parry Moon, The Abacus: Its History, Its Design, Its Possibilities in the Modern World, D. Gordon & Breach Science Pub., 179pp. (1971) (ISBN 978-0677019604).
Parry Moon & Domina Eberle Spencer, The Photic Field, MIT Press, 267pp. (1981) (ISBN 978-0262131667).
Parry Moon & Domina Eberle Spencer, Theory of Holors, Cambridge University Press, 392pp. (1986) (ISBN 978-0521245852).^[1]

Papers

Parry Moon & Domina Eberle Spencer, "Binary Stars and the Velocity of Light", Journal of the Optical Society of America, V43, pp. 635–641 (1953).
Parry Moon & Domina Eberle Spencer, "Electromagnetism Without Magnetism: An Historical Approach", American Journal of Physics, V22, N3, pp. 120–124 (Mar 1954).
Parry Moon & Domina Eberle Spencer, "Interpretation of the Ampere Force", Journal of the Franklin Institute, V257, pp. 203–220 (1954).
Parry Moon & Domina Eberle Spencer, "The Coulomb Force and the Ampere Force, Journal of the Franklin Institute, V257, pp. 305-315 (1954).
Parry Moon & Domina Eberle Spencer, "A New Electrodynamics", Journal of the Franklin Institute, V257, pp. 369–382 (1954).
Parry Moon & Domina Eberle Spencer, "A Postulational Approach to Electromagnetism", Journal of the Franklin Institute, V259, pp. 293–305 (1955).
Parry Moon & Domina Eberle Spencer, "On Electromagnetic Induction", Journal of the Franklin Institute, V260, pp. 213–226 (1955).
Parry Moon & Domina Eberle Spencer, "On the Ampere Force", Journal of the Franklin Institute, V260, pp. 295–311 (1955).
Parry Moon & Domina Eberle Spencer, "Some Electromagnetic Paradoxes", Journal of the Franklin Institute, V260, pp. 373–395 (1955).
Parry Moon & Domina Eberle Spencer, "On the Establishment of Universal Time", Philosophy of Science, V23, pp. 216–229 (1956).
Parry Moon & Domina Eberle Spencer, "The Cosmological Principle and the Cosmological Constant", Journal of the Franklin Institute, V266, pp. 47–58 (1958).
Parry Moon & Domina Eberle Spencer, "Retardation in Cosmology", Philosophy of Science, V25, pp. 287–292 (1958).
Parry Moon & Domina Eberle Spencer, "Mach’s Principle", Philosophy of Science, V26, pp. 125–134 (1958).

References

^ ^a ^b ^c ^d ^e Moon, Parry Hiram; Spencer, Domina Eberle (1986). Theory of Holors: A Generalization of Tensors. Cambridge University Press. ISBN 978-0-521-01900-2.
^ ^a ^b Parry Moon & Domina Eberle Spencer, Foundations of Electrodynamics, D. Van Nostrand Co., 314pp. (1960) (ASIN B000OET7UQ).
^ Moon, Parry Hiram; Spencer, Domina Eberle (1965). Vectors. D. Van Nostrand Co.
^ Spencer, Domina Eberle; Moon, Parry Hiram (1974), "A Unified Approach to Hypernumbers", in Cohen, Robert S.; Stachel, J.J.; Wartofsky, Marx W. (eds.), For Dirk Struik: Scientific, Historical and Political Essays in Honor of Dirk J. Struik, Boston Studies in the Philosophy of Science, vol. 15, Springer, Dordrecht, pp. 101–119, ISBN 978-90-277-0379-8
^ /ˈmiːreɪts/; Greek μέρος "part"
^ Fijalkowski, B.T. (2016). Mechatronics: Dynamical systems approach and theory of holors. IOP Publishing Ltd. ISBN 978-0-7503-1351-3.
^ Rivard, G. (June 1977). "Direct fast Fourier transform of bivariate functions". IEEE Transactions on Acoustics, Speech, and Signal Processing ( Volume: 25, Issue: 3, Jun 1977 ). 25 (3). IEEE: 250–252. doi:10.1109/TASSP.1977.1162951. ISSN 0096-3518.
^ Baciu, G.; Kunii, T.L. (19–24 June 2000). "Homological invariants and holorgraphic representations of topological structures in cellular spaces". Proceedings Computer Graphics International 2000. Geneva, Switzerland, Switzerland: IEEE. doi:10.1109/CGI.2000.852324. ISBN 0-7695-0643-7.{{cite conference}}: CS1 maint: date format (link)
^ German: Valenz; originally introduced to differential geometry by Jan Arnoldus Schouten and Dirk Jan Struik in their 1935 Einführung in die neueren Methoden der Differentialgeometrie. In that work, they explain that they chose the term 'valence' in order to dissolve the confusion created by the use of ambiguous terms such as 'grade', Grad (not to be confused with the concept of grade in geometric algebra), or 'order', Ordnung, for the concept of (tensor) order/degree/rank (not to be confused with the concept of the rank of a tensor in the context of generalizations of matrix rank). (Schouten and Struik, Einführung in die neueren methoden der differentialgeometrie, vol. 1, Noordhoff, 1935, p. 7). Cf. Moon and Spencer, Theory of Holors, p. 12.
^ /ˈplɛθɒs/; Greek: πλῆθος "multitude" or "magnitude, size, extent, amount, quantity", here in the sense of "dimensionality (of a vector)".
^ /eɪˈkɪnətər/; Greek ἀκίνητος "fixed", here in the sense of "invariant".
^ /ˈuːdər/; Greek οὐ "not".
^ Greek ῥάβδος "rod".
^ A vector whose direction and line of application are prescribed, but whose point of application is not prescribed.
^ Greek κινέω "to move"
^ Greek στροφή "a turning"
^ Greek ἑλίσσω "to roll, to wind round".

[holors-1] Moon, Parry Hiram; Spencer, Domina Eberle (1986). Theory of Holors: A Generalization of Tensors. Cambridge University Press. ISBN 978-0-521-01900-2.

[foundations-2] Parry Moon & Domina Eberle Spencer, Foundations of Electrodynamics, D. Van Nostrand Co., 314pp. (1960) (ASIN B000OET7UQ).

[3] Moon, Parry Hiram; Spencer, Domina Eberle (1965). Vectors. D. Van Nostrand Co.

[4] Spencer, Domina Eberle; Moon, Parry Hiram (1974), "A Unified Approach to Hypernumbers", in Cohen, Robert S.; Stachel, J.J.; Wartofsky, Marx W. (eds.), For Dirk Struik: Scientific, Historical and Political Essays in Honor of Dirk J. Struik, Boston Studies in the Philosophy of Science, vol. 15, Springer, Dordrecht, pp. 101–119, ISBN 978-90-277-0379-8

[5] /ˈmiːreɪts/; Greek μέρος "part"

[6] Fijalkowski, B.T. (2016). Mechatronics: Dynamical systems approach and theory of holors. IOP Publishing Ltd. ISBN 978-0-7503-1351-3.

[7] Rivard, G. (June 1977). "Direct fast Fourier transform of bivariate functions". IEEE Transactions on Acoustics, Speech, and Signal Processing ( Volume: 25, Issue: 3, Jun 1977 ). 25 (3). IEEE: 250–252. doi:10.1109/TASSP.1977.1162951. ISSN 0096-3518.

[8] Baciu, G.; Kunii, T.L. (19–24 June 2000). "Homological invariants and holorgraphic representations of topological structures in cellular spaces". Proceedings Computer Graphics International 2000. Geneva, Switzerland, Switzerland: IEEE. doi:10.1109/CGI.2000.852324. ISBN 0-7695-0643-7.{{cite conference}}: CS1 maint: date format (link)

[9] German: Valenz; originally introduced to differential geometry by Jan Arnoldus Schouten and Dirk Jan Struik in their 1935 Einführung in die neueren Methoden der Differentialgeometrie. In that work, they explain that they chose the term 'valence' in order to dissolve the confusion created by the use of ambiguous terms such as 'grade', Grad (not to be confused with the concept of grade in geometric algebra), or 'order', Ordnung, for the concept of (tensor) order/degree/rank (not to be confused with the concept of the rank of a tensor in the context of generalizations of matrix rank). (Schouten and Struik, Einführung in die neueren methoden der differentialgeometrie, vol. 1, Noordhoff, 1935, p. 7). Cf. Moon and Spencer, Theory of Holors, p. 12.

[10] /ˈplɛθɒs/; Greek: πλῆθος "multitude" or "magnitude, size, extent, amount, quantity", here in the sense of "dimensionality (of a vector)".

[11] /eɪˈkɪnətər/; Greek ἀκίνητος "fixed", here in the sense of "invariant".

[12] /ˈuːdər/; Greek οὐ "not".

[13] Greek ῥάβδος "rod".

[14] A vector whose direction and line of application are prescribed, but whose point of application is not prescribed.

[15] Greek κινέω "to move"

[16] Greek στροφή "a turning"

[17] Greek ἑλίσσω "to roll, to wind round".

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