Axioms for Plane Geometry

This chapter shows that the counterintuitive aspects of special relativity are due to the geometry of spacetime. We begin by showing, in the familiar context of plane geometry, how a metric equation separates frame‐dependent quantities from invariant ones. The components of a displacement vector depend on the coordinate system you choose, but its magnitude (the distance between two points, which is more physically meaningful) is invariant. Similarly, space and time components of a spacetime displacement are frame‐dependent, but the magnitude (proper time) is invariant and more physically meaningful. In plane geometry displacements in both x and y contribute positively to the distance, but in spacetime geometry the spatial displacement contributes negatively to the proper time. This is the source of counterintuitive aspects of special relativity. We develop spacetime intuition by practicing with a graphic stretching‐triangle representation of spacetime displacement vectors.

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Three-point perspective and plane geometry

Journal of Mathematics and the Arts ◽

10.1080/17513470701884180 ◽

2007 ◽

Vol 1 (4) ◽

pp. 213-223 ◽

Cited By ~ 1

Author(s):

Marc Frantz ◽

Annalisa Crannell

Keyword(s):

Plane Geometry

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Static and dynamic response of sandwich beams with lattice and pantographic cores

Journal of Sandwich Structures & Materials ◽

10.1177/10996362211033896 ◽

2021 ◽

pp. 109963622110338

Author(s):

Yury Solyaev ◽

Arseniy Babaytsev ◽

Anastasia Ustenko ◽

Andrey Ripetskiy ◽

Alexander Volkov

Keyword(s):

Mechanical Performance ◽

Lattice Structure ◽

Unit Length ◽

Experimental Tests ◽

Plane Geometry ◽

Sandwich Beams ◽

Static Tests ◽

Three Point Bending ◽

The Core ◽

The Impact

Mechanical performance of 3d-printed polyamide sandwich beams with different type of the lattice cores is investigated. Four variants of the beams are considered, which differ in the type of connections between the elements in the lattice structure of the core. We consider the pantographic-type lattices formed by the two families of inclined beams placed with small offset and connected by stiff joints (variant 1), by hinges (variant 2) and made without joints (variant 3). The fourth type of the core has the standard plane geometry formed by the intersected beams lying in the same plane (variant 4). Experimental tests were performed for the localized indentation loading according to the three-point bending scheme with small span-to-thickness ratio. From the experiments we found that the plane geometry of variant 4 has the highest rigidity and the highest load bearing capacity in the static tests. However, other three variants of the pantographic-type cores (1–3) demonstrate the better performance under the impact loading. The impact strength of such structures are in 3.5–5 times higher than those one of variant 4 with almost the same mass per unit length. This result is validated by using numerical simulations and explained by the decrease of the stress concentration and the stress state triaxiality and also by the delocalization effects that arise in the pantographic-type cores.

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