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The hypercubes are one of the few families of regular polytopes that are represented in any number of dimensions. [8] The hypercube (offset) family is one of three regular polytope families, labeled by Coxeter as γ n. The other two are the hypercube dual family, the cross-polytopes, labeled as β n, and the simplices, labeled as α n.
The two general forms of the hypercube honeycombs are the regular form with identical hypercubic facets and one semiregular, with alternating hypercube facets, like a checkerboard. A third form is generated by an expansion operation applied to the regular form, creating facets in place of all lower-dimensional elements.
The relationship between Hammond's postulate and the BEP principle can be understood by considering a S N 1 reaction. Although two transition states occur during a S N 1 reaction (dissociation of the leaving group and then attack by the nucleophile), the dissociation of the leaving group is almost always the rate-determining step. Hence, the ...
Alternation of the n-cube yields one of two n-demicubes, as in this 3-dimensional illustration of the two tetrahedra that arise as the 3-demicubes of the 3-cube.. In geometry, demihypercubes (also called n-demicubes, n-hemicubes, and half measure polytopes) are a class of n-polytopes constructed from alternation of an n-hypercube, labeled as hγ n for being half of the hypercube family, γ n.
In polymer chemistry, polymerization (American English), or polymerisation (British English), is a process of reacting monomer molecules together in a chemical reaction to form polymer chains or three-dimensional networks. [1] [2] [3] There are many forms of polymerization [4] and different systems exist to categorize them. IUPAC definition for ...
A primary kinetic isotope effect (PKIE) may be found when a bond to the isotopically labeled atom is being formed or broken. [3] [4]: 427 Depending on the way a KIE is probed (parallel measurement of rates vs. intermolecular competition vs. intramolecular competition), the observation of a PKIE is indicative of breaking/forming a bond to the isotope at the rate-limiting step, or subsequent ...
In this tiling of the plane by congruent squares, the green and violet squares meet edge-to-edge as do the blue and orange squares. In geometry, Keller's conjecture is the conjecture that in any tiling of n-dimensional Euclidean space by identical hypercubes, there are two hypercubes that share an entire (n − 1)-dimensional face with each other.
In graph theory, the hypercube graph Q n is the graph formed from the vertices and edges of an n-dimensional hypercube. For instance, the cube graph Q 3 is the graph formed by the 8 vertices and 12 edges of a three-dimensional cube. Q n has 2 n vertices, 2 n – 1 n edges, and is a regular graph with n edges touching each vertex.