Categories of intuitive reasoning and GeoGebra 3D: an experience with Brazilian students

This work presents the result of the application of a didactic sequence designed to understand the concept of the Cavalieri’s Principle, supported by the GeoGebra application in its version for mobile phones - 3D Calculator. For this study, the Theory of Categories of Intuitive Reasoning, by Efraim Fischbein, was used as a conceptual basis. The objective of this work was to elaborate and develop a didactic sequence aiming to subsidize the learning of the Cavalieri’s Principle from GeoGebra, as a way to help the student in the construction of geometric reasoning, through visualization, perception and intuition. The methodology of this work is qualitative research, exploratory type, being carried out from a didactic sequence developed in two meetings remotely, due to the scenario of the COVID-19 pandemic. The target audience of this research is a group of students aged 15-17 years from a public school in Fortaleza - CE, Brazil. In summary, it is pointed out that the intuitive reasoning categories mobilized from the use of GeoGebra have great potential to stimulate the evolution of the student's geometric thinking, through the development of perception, intuition and geometric visualization.

Comparison of volume of the forebrain, subarachnoid space and lateral ventricles between dogs with idiopathic epilepsy and controls using a stereological approach: Cavalieri’s principle

Background Canine idiopathic epilepsy (IE) is the most common chronic neurological brain disease in dogs, yet it can only be diagnosed by exclusion of all other potential causes. In people, epilepsy has been associated with a reduction in brain volume. The objective was to estimate the volume of the forebrain (FB), subarachnoid space (SAS) and lateral ventricles (LV) in dogs with IE compared to controls using Cavalieri’s principle. MRI scans of case and control dogs were identified from two neurology referral hospital databases. Eight breeds with increased odds of having IE were included: Golden Retriever, Labrador Retriever, Cocker Spaniel, Border terrier, German Shepherd dog, Parson Jack Russell terrier, Boxer, and Border Collie. Five dogs of each breed with IE and up to five controls were systematically and uniformly randomly sampled (SURS). The volume of the FB, SAS and LV were estimated from MRI scans by one blinded observer using Cavalieri’s principle. Results One hundred-two dogs were identified; 56 were diagnosed with IE and 46 were controls. There was no statistically significant difference in FB, SAS and LV volume between dogs with IE and controls. Dogs with a history of status epilepticus had significantly larger FB than those without (p = 0.05). There was a border-line trend for LV volume to increase with increasing length of seizure history in the IE group (p = 0.055). Conclusion The volumes of the FB, SAS and LV are not different between dogs with IE and controls, so IE remains a diagnosis of exclusion with no specific neuroanatomical biomarkers identified. This is the first time FB and SAS volume has been compared in dogs with IE. Unfortunately, we have shown that the results reporting significantly larger FBs in dogs with status epilepticus and LV volume increase with length of seizure history were likely confounded by breed and should be interpreted cautiously. Whilst these associations are interesting and clinically relevant, further investigation with breed-specific or larger, breed-diverse populations are required to permit strong conclusions. The Cavalieri principle provided an effective estimation of FB, SAS and LV volumes on MRI, but may be too time-intensive for use in clinical practice.

Area-Preserving Transformations: Cavalieri in 2D

A 2D version of Cavalieri's Principle is productive for the teaching of area. In this manuscript, we consider an area-preserving transformation, “segment-skewing,” which provides alternative justification methods for area formulas, conceptual insights into statements about area, and foreshadows transitions about area in calculus via the Riemann integral.

ON THE VOLUME FROM PLANAR SECTIONS THROUGH A CURVE

We derive a formula to obtain the volume of a compact domain from planar sections through a curve. From this formula we propose a stereological estimator for the volume which generalizes some known unbiased estimators which use a systematic sampling scheme. Moreover we formulate a Cavalieri's principle for compact domains is spaces of constant curvature λ.

Le problème du continu pour la mathématisation galiléenne et la géométrie cavalierienne

What reasons can a physicist have to reject the principle of a mathematical method, which he nonetheless uses (even in an implicit way) and which he used frequently in his unpublished works? We are concerned here with Galileo’s doubts and objections against Cavalieri’s “geometry of indivisibles.” One may be astonished by Galileo’s behaviour: Cavalieri’s principle is implied by the Galilean mathematization of naturally accelerated motion; some Galilean demonstrations in fact hinge on it. Yet, in the Discorsi (1638) Galileo seems to be opposed to this principle. e fundamental reason of Galileo’s reluctance with respect to Cavalieri’s geometry is to be sought in Galileo’s ideal of intelligibility. It is true that Galilean physics, and more particularly Galileo’s theories of motion and matter, faces deep paradoxes, which Cavalieri’s geometry succeeds to avoid, thanks to a clear determination of the concept of “aggregatum.” But while avoiding these difficulties, Cavalieri does not furnish any solution for the problems raised by Galilean physics.