On relatively short and long sides of convex pentagons

By the relative distance of pointsa andb of a convex bodyC we mean the ratio of the Euclidean distance ofa andb to the half of the Euclidean distance ofa, b C such thatab is a longest chord ofC parallel to the segmentab. We say that a sideab of a convexn-gon is relatively short (respectively: relatively long) if the relative distance ofa andb is at most (respectively: at least) the relative distance of two consecutive vertices of the regularn-gon. We show that every convexn-gon, wheren 5, has a relatively short side and a relatively long side, and that it is affine-regular if and only if all its sides are of equal relative lengths.Research supported in part by Komitet Bada Naukowych (Committee of Scientific Research), grant number 2 2005 92 03.     

Using dynamic graphs to reveal student reasoning

Using dynamic graphs, future secondary mathematics teachers were able to represent and communicate their understanding of a brief mathematical investigation in a way that a symbolic proof of the problem could not. Four different student work samples are discussed.     

Large triangles contained in the unit disk of a Minkowski plane

We prove that the unit disk C of an arbitrary Minkowski plane contains an equilateral triangle in at least one of the orientations, whose oriented side lengths are ${\frac{3}{2}}$ . We also prove that C permits to inscribe a triangle whose sides are of lengths at least ${\frac{3}{2}}$ in the positive orientation, or that they are of lengths at least ${\frac{3}{2}}$ in the negative orientation. The ratio ${\frac{3}{2}}$ in both the theorems is best possible.     

Estimates for the minimal width of polytopes inscribed in convex bodies

The paper deals with the problem of approximating point sets byn-point subsets with respect to the minimal widthw. Let, in particular, ^d denote the family of all convex bodies in Euclideand-space, letA ^d and letn be an integer greater thand. Then we ask for the greatest number = _n (A) such that everyA A contains a polytope withn vertices which has minimal width at least w(A). We give bounds for _n ( ^d ), for _n (2133; ^d ), and for _n (W ^d ), where 2133; ^d ,W ^d denote the families of centrally symmetric convex bodies and of bodies of constant width, respectively.Dedicated to Professor L. danzer on the occasion of his sixtieth birthdayResearch for this paper was conducted in the academic year 1986/87 while both authors were visiting the University of Washington, Seattle. P. Gritzmann was supported by the Alexander von Humboldt Foundation.     

Essential oil composition of three Australian endemic species of Darwinia (Myrtaceae)

The essential oils from the aerial parts of three rare Australian endemic species of Darwinia have been extracted by hydrodistillation and analyzed by gas chromatography (GC) and gas chromatography coupled to mass spectrometry (GC-MS). In D. procera, myrtenyl acetate (6.1-29.6%), alpha-pinene (6.9-25.1%), gamma-terpinene (6.2-13.6%), bicyclogermacrene (5.5-10.8%) and (E)-nerolidol (3.4-9.7%) were the principal components detected. D. fascicularis ssp. fascicularis produced an oil in which (E)-nerolidol (33.0%), alpha-pinene (15.1%) and gamma-terpinene (10.2%) were the principal components. In D. peduncularis the major constituents were alpha-pinene (33.5%), gamma-terpinene (23.1%) and bicyclogermacrene (6.7%).     

Benetzungswärmen dotierter Festkörper

Summary Heats of immersion have been measured by the method ofStowe with butyl derivatives of different dipol moments as liquids and with doped semiconductingp- andn-alumina, ion charged zeolithes and with cracking catalysts as solids. The heat of immersion is proportional to the dipol moment of the liquid and can be used as measure of the polarity of the surface. In doped aluminas it has a maximum in the undoped (intrinsic) samples, which shows that a dope whatsoever removes dipols from the surface. In zeolithes substitution of sodium by divalent cations leads to a decrease of the immersion heat, parallel to the activation energy of the cumene cracking. The effect is interpreted by cross linking and hence closing of pores for butyl compounds and cumene. This is confirmed by the absence of the effect with the smaller molecule of methanol.

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Vorgetragen auf dem XX. IUPAC-Kongre\, Moskau 1965.

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Dem Fond der Chemischen Industrie und der Deutschen Forschungsgemeinschaft haben wir für Unterstützung der Untersuchung zu danken.