du·al·i·ty

– noun
1. a dual state or quality.
2. a symmetry within a mathematical system such that a theorem remains valid if certain objects, relations, or operations are interchanged, as the interchange of points and lines in a plane in projective geometry.

sym·me·try

– noun
1. the correspondence in size, form, and arrangement of parts on opposite sides of a plane, line, or point; regularity of form or arrangement in terms of like, reciprocal, or corresponding parts.
2. the proper or due proportion of the parts of a body or whole to one another with regard to size and form; excellence of proportion.
3. beauty based on or characterized by such excellence of proportion.
4. a. a geometrical or other regularity that is possessed by a mathematical object and is characterized by the operations that leave the object invariant.
b. a rotation or translation of a plane figure that leaves the figure unchanged although its position may be altered.
5. a property of a physical system that is unaffected by certain mathematical transformations as, for example, the work done by gravity on an object, which is not affected by any change in the position from which the potential energy of the object is measured.

u·ni·ty

– noun
1. the state of being one; oneness.
2. a whole or totality as combining all its parts into one.
3. the state or fact of being united or combined into one, as of the parts of a whole; unification.
4. absence of diversity; unvaried or uniform character.
5. oneness of mind, feeling, etc., as among a number of persons; concord, harmony, or agreement.
6. a. the number one; a quantity regarded as one.
b. identity.
7. a relation of all the parts or elements of a work constituting a harmonious whole and producing a single general effect.

i·den·ti·ty

– noun
1. the state or fact of remaining the same one or ones, as under varying aspects or conditions.
2. the condition of being oneself or itself, and not another.
3. condition or character as to who a person or what a thing is.
4. the state or fact of being the same one as described.
5. the sense of self, providing sameness and continuity in personality over time and sometimes disturbed in mental illnesses, as schizophrenia.
6. exact likeness in nature or qualities.
7. an instance or point of sameness or likeness.
8. an assertion that two terms refer to the same thing.
9. a. an equation that is valid for all values of its variables.
b. an element in a set such that the element operating on any other element of the set leaves the second element unchanged.
c. the property of a function or map such that each element is mapped into itself.
d. the function or map itself.
10. an interesting, famous, or eccentric resident, usually of long standing in a community.