Euclidean rigor and the curious case of the (missing) reflex angle.

I examine a known case of undergeneralization in Euclid's Elements arising from Euclid's non-cognizance of the reflex angle. Sir Thomas Heath (1956) attributed the undergeneralization to Euclid's lack of awareness, an assessment that I dispute. Non-recognition of reflex angles also leads to Proclus' four-sided triangles which violate an essential property of triangles. I show that these issues are resolvable. However, the question as to why Euclid did not acknowledge the reflex angle remains. I claim that the best explanation is that Euclid was guided by reasons of rigor. I argue that the propositional role of diagrammata in Greek mathematics as expounded by Netz (1998,2003) and Euclid's view of the reliability of sense perception impose a crucial representational constraint that governs how diagrams could be used in the Elements. I show that the missing reflex angle, and the careful crafting of definitions in the Elements are evidence that the representation constraint was indeed meticulously followed, even at apparent cost. I argue that alternative explanations for the missing reflex angle are not tenable. In sum, Euclid was aware of the limitations of diagrams, and worked assiduously within their limitations to preserve rigor.

Understanding Higher-Order Interactions in Information Space.

Methods used in topological data analysis naturally capture higher-order interactions in point cloud data embedded in a metric space. This methodology was recently extended to data living in an information space, by which we mean a space measured with an information theoretical distance. One such setting is a finite collection of discrete probability distributions embedded in the probability simplex measured with the relative entropy (Kullback-Leibler divergence). More generally, one can work with a Bregman divergence parameterized by a different notion of entropy. While theoretical algorithms exist for this setup, there is a paucity of implementations for exploring and comparing geometric-topological properties of various information spaces. The interest of this work is therefore twofold. First, we propose the first robust algorithms and software for geometric and topological data analysis in information space. Perhaps surprisingly, despite working with Bregman divergences, our design reuses robust libraries for the Euclidean case. Second, using the new software, we take the first steps towards understanding the geometric-topological structure of these spaces. In particular, we compare them with the more familiar spaces equipped with the Euclidean and Fisher metrics.

From Magnitudes to Geometry and Back: De Zolt's Postulate.

A crucial trend of nineteenth-century mathematics was the search for pure foundations of specific mathematical domains by avoiding the obscure concept of magnitude. In this paper, we examine this trend by considering the "fundamental theorem" of the theory of plane area: "If a polygon is decomposed into polygonal parts in any given way, then the union of all but one of these parts is not equivalent to the given polygon." This proposition, known as De Zolt's postulate, was conceived as a strictly geometrical expression of the general principle of magnitudes "the whole is greater than the part." On the one hand, we illustrate this striving for purity in the foundations of geometry by analysing David Hilbert's classical proof of De Zolt's postulate. On the other hand, we connect this geometrical problem with the first axiomatizations of the concept of magnitude by the end of the nineteenth century. In particular, we argue that a recent result in the logical analysis of the concept of magnitude casts new light on Hilbert's proof. We also outline an alternative development of a theory of magnitude that includes a proof of De Zolt's postulate in an abstract setting.

Ontogenetic Changes in Body Shape and the Scaling of Metabolic Rate in the American Eel (Anguilla rostrata).

AbstractThe body mass (M) scaling of resting metabolic rate (RMR) may vary significantly throughout ontogeny for multiple reasons that are not perfectly understood. To compare two major geometric theories of metabolic scaling, surface area (SA) theory and resource transport network (RTN) theory, we tested whether ontogenetic shifts in metabolic scaling relate to changes in body shape in the American eel (Anguilla rostrata). To do so, we compared the log-linear scaling exponents of RMR to M (bR) and M to body length (bL) in juvenile and subadult eels (glass and yellow eel life stages, respectively). Glass eels exhibited a bL>3 and bR significantly <2/3, as predicted by SA theory. Yellow eels also had a bL>3, but their bR was not significantly different from 2/3 or 3/4. We hypothesize that two developmental changes contribute to bR being higher for yellow eels: (1) a greater reliance on branchial respiration than body-surface-dependent cutaneous respiration and (2) a lower rate of thickening during subadult growth. An ontogenetic decrease in the ratio of cutaneous to gill respiration may have increased the relative importance of the physical constraints of a single-pump, closed circulatory system on the body-size-dependent rate of resource supply to metabolizing tissues (as predicted by RTN theory) in subadult eels. Future research is needed to quantify these developmental changes and their potential mechanistic effects on metabolic scaling, especially in the elver, a critical life stage between the glass and yellow eel stages, that was not analyzed in this study.

Visual foundations of Euclidean geometry.

Geometry defines entities that can be physically realized in space, and our knowledge of abstract geometry may therefore stem from our representations of the physical world. Here, we focus on Euclidean geometry, the geometry historically regarded as "natural". We examine whether humans possess representations describing visual forms in the same way as Euclidean geometry - i.e., in terms of their shape and size. One hundred and twelve participants from the U.S. (age 3-34 years), and 25 participants from the Amazon (age 5-67 years) were asked to locate geometric deviants in panels of 6 forms of variable orientation. Participants of all ages and from both cultures detected deviant forms defined in terms of shape or size, while only U.S. adults drew distinctions between mirror images (i.e. forms differing in "sense"). Moreover, irrelevant variations of sense did not disrupt the detection of a shape or size deviant, while irrelevant variations of shape or size did. At all ages and in both cultures, participants thus retained the same properties as Euclidean geometry in their analysis of visual forms, even in the absence of formal instruction in geometry. These findings show that representations of planar visual forms provide core intuitions on which humans' knowledge in Euclidean geometry could possibly be grounded.

Euclid's Random Walk: Developmental Changes in the Use of Simulation for Geometric Reasoning.

Euclidean geometry has formed the foundation of architecture, science, and technology for millennia, yet the development of human's intuitive reasoning about Euclidean geometry is not well understood. The present study explores the cognitive processes and representations that support the development of humans' intuitive reasoning about Euclidean geometry. One-hundred-twenty-five 7- to 12-year-old children and 30 adults completed a localization task in which they visually extrapolated missing parts of fragmented planar triangles and a reasoning task in which they answered verbal questions about the general properties of planar triangles. While basic Euclidean principles guided even young children's visual extrapolations, only older children and adults reasoned about triangles in ways that were consistent with Euclidean geometry. Moreover, a relation beteen visual extrapolation and reasoning appeared only in older children and adults. Reasoning consistent with Euclidean geometry may thus emerge when children abandon incorrect, axiomatic-based reasoning strategies and come to reason using mental simulations of visual extrapolations.

Human navigation in curved spaces.

Navigation and representations of the spatial environment are central to human survival. It has often been debated whether spatial representations follow Euclidean principles, and a number of studies challenged the Euclidean hypothesis. Two experiments examined the geometry of human navigation system using true non-Euclidean environments, i.e., curved spaces with non-Euclidean geometry at every point of the space. Participants walked along two legs in an outbound journey, then pointed to the direction of the starting point (home). The homing behavior was examined in three virtual environments, Euclidean space, hyperbolic space, and spherical space. The results showed that people's responses matched the direction of Euclidean origin, regardless of the curvature of the space itself. Moreover, participants still responded as if the space were Euclidean when a learning period was added for them to explore the spatial properties of the environment before performing the homing task to ensure violations of Euclidean geometry were readily detected. These data suggest that the path integration / spatial updating system operates on Euclidean geometry, even when curvature violations are clearly present.

Bimanual thumb-index finger indications of noncorresponding extents.

Two experiments tested a prediction derived from the recent finding that the Oppel-Kundt illusion - the overestimation of a filled extent relative to an empty one - was much attenuated when the empty part of a bipartite row of dots was vertical and the filled part horizontal, suggesting that the Horizontal-vertical illusion - the overestimation of vertical extents relative to horizontal ones - only acted on the empty part of an Oppel-Kundt figure. Observers had to bimanually indicate the sizes of the two parts of an Oppel-Kundt figure, which were arranged one above the other with one part vertical and the other part tilted -45°, 0°, or 45°. Results conformed to the prediction but response bias was greater when observers had been instructed to point to the extents' endpoints than when instructed to estimate the extents' lengths, suggesting that different concepts and motor programs had been activated.

Quadratic Split Quaternion Polynomials: Factorization and Geometry.

We investigate factorizability of a quadratic split quaternion polynomial. In addition to inequality conditions for existence of such factorization, we provide lucid geometric interpretations in the projective space over the split quaternions.

Compositional data analysis in epidemiology.

Compositional data analysis refers to analyzing relative information, based on ratios between the variables in a data set. Data from epidemiology are usually treated as absolute information in an analysis. We outline the differences in both approaches for univariate and multivariate statistical analyses, using illustrative data sets from Austrian districts. Not only the results of the analyses can differ, but in particular the interpretation differs. It is demonstrated that the compositional data analysis approach leads to new and interesting insights.

A new method for evaluating the normal rake angle and inclination angle on medical needles.

Hollow needles are the most frequently used medical equipment. The design of a hollow needle that best enables medical procedures requires a better understanding of needle tip geometry. Calculating the cutting angles of a needle for a complex surface topology is difficult. This article proposes a new method based on non-Euclidean geometry for the analysis of biopsy needle tip. The method can be used to calculate the cutting angles on any pipe needle. To verify the validity of this method, the normal rake angle and inclination angle on four types of needles (bias bevel needle, cylinder surface needle, curved surface needle and Cournand-type needle) were investigated. It was found that calculation of the cutting angles was simple and convenient using this method, especially for the curved surface needles. Images of the cutting angles from the Cournand-type needles revealed that the smaller bevel angle [Formula: see text] resulted in a higher normal rake angle [Formula: see text] and inclination angle [Formula: see text]. As [Formula: see text] increased, the range of the normal rake angle [Formula: see text] became larger at first and then became smaller.

The extent of visual space inferred from perspective angles.

Retinal images are perspective projections of the visual environment. Perspective projections do not explain why we perceive perspective in 3-D space. Analysis of underlying spatial transformations shows that visual space is a perspective transformation of physical space if parallel lines in physical space vanish at finite distance in visual space. Perspective angles, i.e., the angle perceived between parallel lines in physical space, were estimated for rails of a straight railway track. Perspective angles were also estimated from pictures taken from the same point of view. Perspective angles between rails ranged from 27% to 83% of their angular size in the retinal image. Perspective angles prescribe the distance of vanishing points of visual space. All computed distances were shorter than 6 m. The shallow depth of a hypothetical space inferred from perspective angles does not match the depth of visual space, as it is perceived. Incongruity between the perceived shape of a railway line on the one hand and the experienced ratio between width and length of the line on the other hand is huge, but apparently so unobtrusive that it has remained unnoticed. The incompatibility between perspective angles and perceived distances casts doubt on evidence for a curved visual space that has been presented in the literature and was obtained from combining judgments of distances and angles with physical positions.

Spatial constant equi-affine speed and motion perception.

The two-thirds power law, postulating an inverse local relation between the velocity and cubed root of curvature of planar trajectories, is a long-established simplifying principle of human hand movements. In perception, the motion of a dot along a planar elliptical path appears most uniform for speed profiles closer to those predicted by the power law than to constant Euclidean speed, a kinetic-visual illusion. Mathematically, complying with this law is equivalent to moving at constant planar equi-affine speed, while unconstrained three-dimensional drawing movements generally follow constant spatial equi-affine speed. Here we test the generalization of this illusion to visual perception of spatial motion for a dot moving along five differently shaped paths, using stereoscopic projection. The movements appeared most uniform for speed profiles closer to constant spatial equi-affine speed than to constant Euclidean speed, with path torsion (i.e., local deviation from planarity) directly affecting the speed profiles perceived as most uniform, as predicted for constant spatial equi-affine speed. This demonstrates the dominance of equi-affine geometry in spatial motion perception. However, constant equi-affine speed did not fully account for the variability among the speed profiles selected as most uniform for different shapes. Moreover, in a followup experiment, we found that viewing distance affected the speed profile reported as most uniform for the extensively studied planar elliptical motion paths. These findings provide evidence for the critical role of equi-affine geometry in spatial motion perception and contribute to the mounting evidence for the role of non-Euclidean geometries in motion perception and production.