The Kissing Triangles: The Aesthetics of Mathematical Discovery

Papert's (1978) appeal to reconsider the power and possibilities of the aesthetic in mathematics learning is often ignored in mathematics education research. This paper begins with the premise, put forth by Dewey (1934), that the aesthetic structures many dimensions of inquiry and experience. In the same way that using particular paintings, musical compositions, or even everyday experiences has been instrumental to attempts by philosophers to understand the aesthetic dimensions of meaning and experience in artistic domains, I propose that analysing a particular encounter with mathematics may help reveal the nature and role of the often nebulous responses of elegance, beauty, and `fit' to which mathematicians lay claim in their mathematical activity. To achieve this, I draw on and adapt the defining features of the aesthetic character of experience set forth by the aesthetician Beardsley (1982). This, in turn, sheds light on the role thataesthetics can play in mathematical inquiry and experience, and provides initial categories and conjectures that can be used to investigate the potential roles of aesthetics in mathematics learning contexts.This revised version was published online in September 2005 with corrections to the Cover Date.     

On removable even circuits in graphs

Let G be a connected graph with minimum degree at least 3. We prove that there exists an even circuit C in G such that G−E(C) is either connected or contains precisely two components one of which is isomorphic to a 1-bond. We further prove sufficient conditions for there to exist an even circuit C in a 2-connected simple graph G such that G−E(C) is 2-connected. As a consequence of this, we obtain sufficient conditions for there to exist an even circuit C in a 2-connected graph G for which G−E(C) is 2-connected.     

Upon the removal of the edges of a 1-factor from an even circuit in a 2-connected graph

Let G be a 2-connected graph with minimum degree at least 3. We prove that there exists an even circuit C in G with factorization F={F₁,F₂} such that G−E(F₁) is 2-connected.     

Investigation of the chemical state of ultrathin Hf-Al-O films during high temperature annealing

Al₂O₃ incorporated HfO₂ films grown by atomic layer deposition (ALD) were investigated by high-resolution X-ray photoelectron spectroscopy (HRXPS). The core level energy state of a 15 Å thick film showed a shift to higher binding energy, as the result of a silicate formation and Al₂O₃ incorporation. The incorporation of Al₂O₃ into the HfO₂ film had no effect on silicate formation at the interface between the film and Si, while the ionic bonding characteristics and hybridization effects were enhanced compared to a pure HfO₂ film. The dissociation of the film in an ultrahigh vacuum (UHV) is effectively suppressed compared to a pure HfO₂ film, indicating an enhanced thermal stability of Hf-Al-O. Any dissociated Al₂O₃ on the film surface was completely removed into the vacuum by vacuum annealing treatment over 850 °C, while HfO₂ contributed to Hf silicide formation on the film surface.     

Enumeration of Cryptosporidium spp. in water with US EPA method 1622, USA

The occurrence of Cryptosporidium parvum or other pathogenic Cryptosporidium species in water must be known in order to assess risk and determine the treatment needed to reduce Cryptosporidium oocysts to acceptable levels in finished drinking water. Because Cryptosporidium oocyst occurrence may be sparse, methods must concentrate a large volume of water and correctly identify oocysts in the concentrate. The U.S. Environmental Protection Agency Information Collection Rule (ICR) protozoan method gives low and variable recoveries of Cryptosporidium oocysts, making risk assessment difficult. Therefore, a method giving better oocyst recovery and more consistent results was needed. Method 1622 was developed with existing materials and procedures, and improvements were made in filtration, cleanup, and detection. Absolute porosity filters were used, with cleanup by immunomagnetic separation and detection by direct fluorescent antibody stain with 4',6-diamidino-2-phenylindole (DAPI) staining for additional cell structures. Both the level and consistency of oocyst recovery were improved compared to recovery with the ICR method.     

Analytical TEM examinations of CoPt-TiO2 perpendicular magnetic recording media.

For this analytical TEM study, nonmagnetic oxygen-rich boundaries were introduced into Co-Pt-alloy perpendicular recording media by cosputtering Co and Pt with TiO2. Increasing the TiO2 content resulted in changes to the microstructure and elemental distribution within grains and boundaries in these films. EFTEM imaging was used to generate composition maps spanning many tens of grains, thereby giving an overall depiction of the changes in elemental distribution occurring with increasing TiO2 content. Comparing EFTEM with spectrum-imaging maps created by high-resolution STEM with EDXS and EELS enabled both corroboration of EFTEM results and quantification of the chemical composition within individual grain boundary areas. The difficulty of interpreting data from EDXS for these extremely thin films is discussed. Increasing the TiO2 content of the media was found to create more uniformly wide Ti- and O-rich grain boundaries as well as Ti- and O-rich regions within grains.