T he values of complete elliptic integrals of the first and the second kind are expressible via power series representations of the hypergeometric function (with corresponding arguments). The complete elliptic integral of the first kind is also known to be eloquently expressible via an arithmetic-geometric mean, whereas (before now) the complete elliptic integral of the second kind has been deprived such an expression (of supreme power and simplicity). With this paper, the quest for a concise formula giving rise to an exact iterative swiftly convergent method permitting the calculation of the perimeter of an ellipse is over! Instead of an IntroductionA recent survey [16] of formulae (approximate and exact) for calculating the perimeter of an ellipse is erroneously resuméd:There is no simple exact formula: There are simple formulas but they are not exact, and there are exact formulas but they are not simple.No breakthrough will be required for a refutation, since most (if not everything!) had long been done by Gauss, merely awaiting a (last) clarification. The Arithmetic-Geometric Mean and a Modification ThereofIntroduce a sequence of pairs {x n , y n } ∞ n=0 : Define the arithmetic-geometric mean (which we shall abbreviate as AGM) of two positive numbers x and y as the (common) limit of the (descending) sequence {x n } ∞ n=1 and the (ascending) sequence {y n } ∞ n=1 with x 0 = x, y 0 = y.

The aim of our paper is to explain a computer animation of the strictly critical rigid body motion, which ought not be confused with any other motion in its "proximity", however close. We demonstrate that the (local) "uniqueness theorem" remarkably fails in the case of critical motion which (time) domain must be compactified via adjoining the point at (complex) infinity. Two (opposite to each other) "flips" correspond to one and the same (initial) rotation, exclusively either clockwise or counterclockwise, (strictly) about the intermediate axis of inertia. These two symmetrical reversals of the direction of the intermediate axis (of inertia), initially matching then opposing the direction of the (fixed) angular momentum, share one and the same (symmetry) axis, which we call "Galois axis". The Galois axis, which is fixed within the body (but coincides with no principal axis of inertia), rotates uniformly in a plane orthogonal to the angular momentum, as our animation demonstrates. The animation also traces the corresponding two (recurrently selfintersecting) herpolhodes, which turn out to be mirror-symmetrical. The "mirror" is exhibited to lie in a plane, orthogonal to Galois axis at the midst of the "flip". The Galois axis itself is reflected across the minor (or the major) axis of inertia if the direction of the angular momentum is reversed. The formula for the "swing" of the intermediate axis in the plane orthogonal to Galois axis (in body's frame), turns out to be "a square root" of Abrarov's critical solution for a simple pendulum, which (imaginary) period is (exactly) calculated.

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This article is devoted to a comparative analysis of the results of the ReMath project (Representing Mathematics with digital media), devoted to the study of digital representations of mathematical concepts. The theoretical provisions and conclusions of this project will be analyzed based on the theory of the information environment [1], developed with the participation of one of the authors of this article. The analysis performed in this work partially coincides with the conclusions of the ReMath project, but uses a different research basis, based mainly on the work of Russian scientists. It is of interest to analyze the work of the ReMath project from the conceptual positions set forth in this monograph and to establish links between concepts and differences in understanding the impact of computer tools (artifacts) on the process of teaching mathematics. At the same time, the authors dispute the interpretation of some issues in Vygotsky’s works by foreign researchers and give their views on the types and functions of digital artifacts in teaching mathematics.

This article is devoted to a comparative analysis of the results of the ReMath project (Representing Mathematics with digital media), devoted to the study of digital representations of mathematical concepts. The theoretical provisions and conclusions of this project will be analyzed based on the theory of the information environment [1], developed with the participation of one of the authors of this article. The analysis performed in this work partially coincides with the conclusions of the ReMath project, but uses a different research basis, based mainly on the work of Russian scientists. It is of interest to analyze the work of the ReMath project from the conceptual positions set forth in this monograph and to establish links between concepts and differences in understanding the impact of computer tools (artifacts) on the process of teaching mathematics. At the same time, the authors dispute the interpretation of some issues in Vygotsky’s works by foreign researchers and give their views on the types and functions of digital artifacts in teaching mathematics.