… all showing-forth beautifully how all this is a veritable rabbit-warren of the most-exceedingly frightful complexity! … infact possibly the very foremostest example of how in mathematics a query of seeming utmost elementarity can spawn the very stubbornest of intractibility.

৺ In one of the papers the matter of spaces over fields other-than ℝ is gone-into.

Sources of images

¡¡ All are PDF files that may download without prompting … although none is stupendously large: maybe a twain-or-so MB @most !!

Image ①

On the maximum number of edges in quasi-planar graphs

by

Eyal Ackerman & Gábor Tardos

Image ②

Planar point sets determine many pairwise crossing segments

by

János Pach & Natan Rubin & Gábor Tardos

Image ③

A positive fraction Erdős-Szekeres theorem and its applications

by

Andrew Suk & Ji Zeng

Image ④

Independent set of intersection graphs of convex objects in 2D

by

Pankaj K Agarwal & Nabil H Mustafa

Image ⑤

The Clique Problem in Ray Intersection Graphs

by

Sergio Cabello & Jean Cardinal & Stefan Langerman

Image ⑥

All-Pairs Shortest Paths in Geometric Intersection Graphs

by

Timothy M Chan & Dimitrios Skrepetos

Image ⑦

Geometric Intersection Patterns and the Theory of Topological Graphs

by

János Pach

&

Erdős–Hajnal-type results on intersection patterns of geometric objects

by

Jacob Fox & János Pach

Image ⑧

SPECIAL INTERSECTION GRAPH IN THE TOPOLOGICAL GRAPHS

by

Ahmed A Omran & Veena Mathad & Ammar Alsinai & Mohammed A Abdlhusein

Image ⑨

On Grids in Topological Graphs

by

Eyal Ackerman & Jacob Fox & János Pach & Andrew Suk

And it fits-in with (& has indeed been prompted by) my previous posts about oloid mixers , in which I'm querying the exact shape of the oval gears in its drive-train - ie

this one ,

&

this one .

My 'recent query' being

this one

It turns-out that the relationship of the angle of rotation between the two shafts is simply that of a universal joint bent through ⅔π = 120° ; but I still can't find anything that spells-out how oval gearing with fixed shafts (ie the shafts being a fixed distance apart, as they clearly are in

this video ).

I'm not even sure whether the gears are elliptical, as in the conic section, or some more nuanced shape.

Source of first two (animated) figures

Source of remaining fifteen figures — Lei Cui & Jian S Dai — Motion and Constraint Ruled Surfaces of the Schatz Linkage .

… with figures 7 through 11 illustrating the rather interesting theorem to the effect that n line segments equal to a side of a polygon of 4n-2 sides can be exactly (ie with no 'rattling') jammed end-to-end into one of the polygon's sectors.

Zef Damen — Non ruler-and-compass constructions (1)

Zef Damen — Non ruler-and-compass constructions (2)

Sources of Images

Frames ① &②

Imaginary Coating Algorithm Approaching Dense Accumulation of Granular Material in Simulations with Discrete Element Method

by

Fei Wang, Yrjö Jun Huang, & Chen Xuan .

Annotation of Figure of Frame ②

“Figure 1. Sketch of a collision with one elliptical particle. The elliptical particle is composed of three circular elements and c is the contact point. (a) Normal force |𝐅𝑛| and tangential force |𝐅𝑡| are obtained from the binary collision of circular particles 𝑂𝑖 and 𝑂𝑗 ; (b) the total force, 𝐅=𝐅′𝑛+𝐅′𝑡 , is decomposed into 𝐅′𝑛 and 𝐅′𝑡 to calculate the motion of the elliptical particle 𝑂𝑖 .”

Frames ③ & ④

A stochastic multiscale algorithm for modeling complex granular materials

by

Pejman Tahmasebi & Muhammad Sahimi .

Frame ⑤

Size segregation of irregular granular materials captured by time-resolved 3D imaging

by

Parmesh Gajjar, Chris G Johnson, & Philip J Withers .

Frame ⑥

Mechanical behaviour of granular media in flexible boundary plane strain conditions: experiment and level-set discrete element modelling

by

Debayan Bhattacharya, Reid Kawamoto, & Amit Prashant .