Frankenheim's focus of research was crystallography, particularly studies of
crystal structure and the mathematical and theoretical basis of the
symmetry of
crystals. By 1826, he was already using the integer
reciprocals of ''Weiss' coefficients'' (the intersection of a plane with the three
crystallographic axes) to describe the spatial positions of crystal surfaces, from which the British crystallographer
William Hallowes Miller (1801-1880) developed the concept of
Miller indices in 1839. By assigning
symmetry elements to the
crystal systems defined previously by Weiss and
Friedrich Mohs (1773-1839), Frankenheim was able, for the first time, to define 32
point groups (
crystal classes) and to classify them into four crystal systems (the regular one, the fourfold, the twofold and the sixfold). He published this result in his 1826 paper "Crystallonomische Aufsätze". Later, Frankenheim derived 15
lattice types for crystals, which were later reduced by
Auguste Bravais (1811-1863) to 14 and today are referred to as
Bravais lattices. On pages 311-312 of his 1835 book Die Lehre von der Cohäsion, Frankenheim says that application of symmetry ideas shows that there are 15 crystal families, but in this book he doesn’t actually describe them. On page 15 of his 1842 treatise System Der Krystalle, however, he says – in reference to what he calls the Grundform (basic shape) of crystals – that “there are a total of fifteen, three of which are tesseral (i.e., cubic) crystals, two are tetragonal, two are hexagonal, four are isoclinic (i.e., orthorhombic), three belong to monoclinic and one to triclinic crystals.” All these are the correct numbers except the three for monoclinic, which should be two. Later, on page 102, he discusses monoclinic crystals in more detail. Frankenheim’s three monoclinic shapes correspond to what we today call primitive, body-centered, and end-centered monoclinic unit cells. The primitive cell describes the primitive monoclinic lattice, but the body-centered and end-centered monoclinic cells describe the same non-primitive lattice: the two different cells can be transformed into one another by a simple redefinition of one of the cell axes. Bravais got the number right (14) in a paper he read to the French Academy of Sciences in 1848 (published in 1850) and also gave a good discussion of why there are exactly 14 lattices. For these reasons we refer to them today as
Bravais lattices. Bravais mentions in a footnote that Frankenheim in his 1842 treatise listed 3 “modes of the oblique prismatic system of Hauy” (i.e., monoclinic crystals). Bravais goes on to say in the footnote that the last two of Frankenheim’s three modes are in fact identical. Interestingly, in 1856 Frankenheim revisited the question in a journal article (“Über die Anordnung der Moleküle im Krystall,” Ann. Phys. Chem. 1856, 97, 337–382. On page 349 he explicitly says that there are only two monoclinic lattices, not three, because the body-centered and end-centered monoclinic possibilities are actually identical by redefining an axis. He says on pages 355-356 that owing to this identity there are 14 families, not 15 as he said in 1835 and 1842. Nowhere in the article does he mention Bravais, however, so possibly he was unaware of Bravais’s 1848 paper. Frankenheim conducted one of the first
microscopic examinations of crystals in
polarized light, using the then-new
Nicol prism as a
polarizer. In the field of geography, his most famous work is his book
Völkerkunde ("Ethnology"), published in 1852. == Publications ==