Holor

A holor (/ˈhoʊlər/; Greek ὅλος "whole") is a mathematical entity that is made up of one or more independent quantities ("merates"^[1] as they are called in the theory of holors). Complex numbers, scalars, vectors, matrices, tensors, quaternions, and other hypercomplex numbers are kinds of holors. 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.^[2]

The term holor was coined by Parry Moon and Domina Eberle Spencer. Moon and Spencer classify holors as either nongeometric objects or geometric objects. They further classify the geometric objects as either oudors or akinetors, where the (contravariant) akinetors^[3] transform as

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

and the oudors^[4] 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.

Holors are furthermore classified with respect to their i) plethos^[5] n, and ii) valence^[6] N.

Moon and Spencer 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^[7] and corresponds to a sliding vector^[8] in standard nomenclature. Other objects in their classification scheme include free rhabdors, kineors,^[9] fixed strophors,^[10] free strophors, and helissors.^[11]

See also

• Affinor
• Geometric vector
• Geometric algebra
• Geometry of matrices, a research program initiated by Hua Luogeng in 1945
• Musean hypernumber

Notes and references

1. /ˈmiːreɪts/; Greek μέρος "part".
2. Moon, Parry Hiram; Spencer, Domina Eberle (1986). Theory of Holors : A Generalization of Tensors. Cambridge University Press. ISBN 978-0-521-01900-2.
3. /eɪˈkɪnətər/; Greek ἀκίνητος "fixed", here in the sense of "invariant".
4. /ˈuːdər/; Greek οὐ "not".
5. /ˈplɛθɒs/; Greek: πλῆθος "multitude", here in the sense of "dimensionality (of a vector)".
6. 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 matrices and tensors or with matrix rank), which is the number of indices needed to label a component of a multi-dimensional array of numerical values). The term 'valence' is to remind the concept of chemical valence (Schouten and Struik 1935, Bd. I, p. 7). See also Moon and Spencer 1989, p. 12.
7. Greek ῥάβδος "rod".
8. A vector whose direction and line of application are prescribed, but whose point of application is not prescribed.
9. Greek κινέω "to move"
10. Greek στροφή "a turning"
11. Greek ἑλίσσω "to roll, to wind round".