From Euclidean Geometry to Knots and Nets

This document is the Accepted Manuscript of an article accepted for publication in Synthese. Under embargo until 19 September 2018. The final publication is available at Springer via https://doi.org/10.1007/s11229-017-1558-x.This paper assumes the success of arguments against the view that informal mathematical proofs secure rational conviction in virtue of their relations with corresponding formal derivations. This assumption entails a need for an alternative account of the logic of informal mathematical proofs. Following examination of case studies by Manders, De Toffoli and Giardino, Leitgeb, Feferman and others, this paper proposes a framework for analysing those informal proofs that appeal to the perception or modification of diagrams or to the inspection or imaginative manipulation of mental models of mathematical phenomena. Proofs relying on diagrams can be rigorous if (a) it is easy to draw a diagram that shares or otherwise indicates the structure of the mathematical object, (b) the information thus displayed is not metrical and (c) it is possible to put the inferences into systematic mathematical relation with other mathematical inferential practices. Proofs that appeal to mental models can be rigorous if the mental models can be externalised as diagrammatic practice that satisfies these three conditions.Peer reviewe

Special theme article : science and sociology of footwear

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Structural foot characteristics in people with midfoot osteoarthritis: Cross‐Sectional findings from the clinical assessment study of the foot

Objective: This study compared radiographic measures of foot structure between people with and without symptomatic radiographic midfoot osteoarthritis (OA). Methods: This was a cross‐sectional study of adults aged 50 years and older registered with four UK general practices who reported foot pain in the past year. Bilateral weightbearing dorsoplantar and lateral radiographs were obtained. Symptomatic radiographic midfoot OA was defined as midfoot pain in the last 4 weeks, combined with radiographic OA in one or more midfoot joints (first cuneometatarsal, second cuneometatarsal, navicular‐first cuneiform, and talonavicular). Midfoot OA cases were matched 1:1 for sex and age to controls with a 5‐year age tolerance. Eleven radiographic measures were extracted and compared between the groups using independent sample t‐tests and effect sizes (Cohen's d). Results: We identified 63 midfoot OA cases (mean ± SD age was 66.8 ± 8.0 years, with 32 male and 31 female participants) and matched these to 63 controls (mean ± SD age was 65.9 ± 7.8 years). There were no differences in metatarsal lengths between the groups. However, those with midfoot OA had a higher calcaneal‐first metatarsal angle (d = 0.43, small effect size, P = 0.018) and lower calcaneal inclination angle (d = 0.46, small effect size, P = 0.011) compared with controls. Conclusions: People with midfoot OA have a flatter foot posture compared with controls. Although caution is required when inferring causation from cross‐sectional data, these findings are consistent with a pathomechanical pathway linking foot structure to the development of midfoot OA. Prospective studies are required to determine the temporal relationships between foot structure, function, and the development of this common and disabling condition

Plantar pressures in people with midfoot osteoarthritis: cross-sectional findings from the Clinical Assessment Study of the Foot

Background Midfoot osteoarthritis (OA) is a common condition, however its aetiology is not well understood. Understanding how plantar pressures differ between people with and without midfoot OA may provide insight into the aetiology and how best to manage this condition. Research question To compare plantar pressures between people with and without symptomatic radiographic midfoot OA. Methods This was a cross-sectional study of adults aged ≥50 years registered with four UK general practices who reported foot pain in the past year. Symptomatic radiographic midfoot OA was defined as midfoot pain in the last four weeks, combined with radiographic OA in one or more midfoot joints. Cases were matched 1:1 for sex and age (± 5 years) to asymptomatic controls. Peak plantar pressure and maximum force in 10 regions of the foot were determined using a pressure platform (RSscan International, Olen, Belgium) and compared between the groups using independent samples t-tests and effect sizes (Cohen’s d). Results We included 61 midfoot OA cases (mean age 67.0, SD 8.1, 31 males, 30 females) and matched these to 61 controls (mean age 66.0, SD 7.9). Midfoot OA cases displayed greater force (d=0.79, medium effect size, p=<0.001) and pressure at the midfoot (d=0.70, medium effect size, p=<0.001), greater force at the fourth metatarsophalangeal (MTP) joint (d=0.28, small effect size, p=0.13), and fifth MTP joint (d=0.37, small effect size, p=0.10) and greater pressure at the fifth MTP joint (d=0.34, small effect size, p=0.13). They also displayed lower force (d=0.40, small effect size, p=0.02) and pressure at the hallux (d=0.50, medium effect size, p=<0.001) and lower force (d=0.54, medium effect size, p=<0.001) and pressure at the lesser toes (d=0.48, small effect size, p=<0.001) compared with controls. Significance Midfoot OA appears associated with lowering of the medial longitudinal arch, greater lateral push off and less propulsion at toe off. Longitudinal studies are needed to establish causal relationships

Soft x ray tomoholography

We demonstrate an x ray imaging method that combines Fourier transform holography with tomography tomoholography for threedimensional 3D microscopic imaging. A 3D image of a diatom shell with a spatial resolution of 140 nm is presented. The experiment is realized by using a small gold sphere as the reference wave source for holographic imaging. This setup allows us to rotate the sample and to collect a number of 2D projections for tomograph

Centre of pressure characteristics in normal, planus and cavus feet

Background The aim of this study was to compare centre of pressure (COP) characteristics between healthy adults with normal, planus or cavus feet who were allocated to groups based on reliable foot posture measurement techniques. Methods Ninety-two healthy adult participants (aged 18 to 45) were recruited and classified as either normal (n = 35), pes planus (n = 31) or pes cavus (n = 26) based on Foot Posture Index, Arch Index and normalised navicular height truncated measurements. Barefoot walking trials were conducted using an emed®-x 400 plantar pressure system (Novel GmbH, Munich, Germany). Average, maximum, minimum and range (difference between maximum and minimum) values were calculated for COP velocity and lateral-medial force index during loading response, midstance, terminal stance and pre-swing phases of stance. The COP excursion index was also calculated. One-way analyses of variance were used to compare the three foot posture groups. Results The cavus foot exhibited the slowest average and minimum COP velocity during terminal stance, but this pattern was reversed during pre-swing, when the cavus foot exhibited the fastest maximum COP velocity. The planus foot exhibited the smallest lateral medial force index range during terminal stance. There were no differences between the groups for COP excursion index. Conclusion These findings indicate that there are differences in COP characteristics between foot postures, which may represent different mechanisms for generating force to facilitate forward progression of the body during the propulsive phases of gait

Movement of the human foot in 100 pain free individuals aged 18–45 : implications for understanding normal foot function

Background: Understanding motion in the normal healthy foot is a prerequisite for understanding the effects of pathology and thereafter setting targets for interventions. Quality foot kinematic data from healthy feet will also assist the development of high quality and research based clinical models of foot biomechanics. To address gaps in the current literature we aimed to describe 3D foot kinematics using a 5 segment foot model in a population of 100 pain free individuals. Methods: Kinematics of the leg, calcaneus, midfoot, medial and lateral forefoot and hallux were measured in 100 self reported healthy and pain free individuals during walking. Descriptive statistics were used to characterise foot movements. Contributions from different foot segments to the total motion in each plane were also derived to explore functional roles of different parts of the foot. Results: Foot segments demonstrated greatest motion in the sagittal plane, but large ranges of movement in all planes. All foot segments demonstrated movement throughout gait, though least motion was observed between the midfoot and calcaneus. There was inconsistent evidence of movement coupling between joints. There were clear differences in motion data compared to foot segment models reported in the literature. Conclusions: The data reveal the foot is a multiarticular structure, movements are complex, show incomplete evidence of coupling, and vary person to person. The data provide a useful reference data set against which future experimental data can be compared and may provide the basis for conceptual models of foot function based on data rather than anecdotal observations

Are clinical measures of foot posture and mobility associated with foot kinematics when walking?

Background: There is uncertainty as to which foot posture measures are the most valid in terms of predicting kinematics of the foot. The aim of this study was to investigate the associations of clinical measures of static foot posture and mobility with foot kinematics during barefoot walking. Method: Foot posture and mobility were measured in 97 healthy adults (46 males, 51 females; mean age 24.4 ± 6.2 years). Foot posture was assessed using the 6-item Foot Posture Index (FPI), Arch Index (AI), Normalised Navicular Height (NNHt) and Normalised Dorsal Arch Height (DAH). Foot mobility was evaluated using the Foot Mobility Magnitude (FMM) measure. Following this, a five-segment foot model was used to measure tri-planar motion of the rearfoot, midfoot, medial forefoot, lateral forefoot and hallux. Peak and range of motion variables during load acceptance and midstance/propulsion phases of gait were extracted for all relative segment to segment motion calculations. Hierarchical regression analyses were conducted, adjusting for potential confounding variables. Results: The degree of variance in peak and range of motion kinematic variables that was independently explained by foot posture measures was as follows: FPI 5 to 22 %, NNHt 6 to 20 %, AI 7 to 13 %, DAH 6 to 8 %, and FMM 8 %. The FPI was retained as a significant predictor across the most number of kinematic variables. However, the amount of variance explained by the FPI for individual kinematic variables did not exceed other measures. Overall, static foot posture measures were more strongly associated with kinematic variables than foot mobility measures and explained more variation in peak variables compared to range of motion variables. Conclusions: Foot posture measures can explain only a small amount of variation in foot kinematics. Static foot posture measures, and in particular the FPI, were more strongly associated with foot kinematics compared with foot mobility measures. These findings suggest that foot kinematics cannot be accurately inferred from clinical observations of foot posture alone