#Crows very good at #geometry, they can not only distinguish the from stars, but choose between distinctive quadrilaterals

https://www.earth.com/news/crows-can-recognize-geometric-patterns-in-shapes/

I have found an interesting geometric fact: suppose you have a hexagon of side 1 and duplicate and enlarge it by the golden ratio 𝜑; the distance from one vertex of the unit hexagon to a vertex of the bigger hexagon 60° apart is √2. Furthermore, if another hexagon reduced by 𝜑 is drawn inside, the distance from one vertex of the unit hexagon to a vertex of the smaller hexagon 120° apart is also √2 [first figure].
This boils down to the fact that a triangle of sides 1, √2, and 𝜑 has an angle of 60° opposite to side √2. That triangle is very remarkable as it contains the three more relevant algebraic geometric constants: √2, √3/2 (altitude to the bigger side) and 𝜑 [second figure]. Of course this can be also used to construct 𝜑 from a square and a triangle (I bet this is known). In the follow-up some artistic designs exploiting those facts.
#geometry #Mathematics #triangle #GoldenRatio

Here's a stupid #geometry / #astronomy question. Planetary orbits are ellipses (under the obvious simplifying assumptions) which are conic sections. So what's the cone? There are infinitely many cones that fit of course. But is there one that best explains? To put it another way, is there a neat geometric argument that an orbit should be an ellipse, that doesn't require too much physics? #mathematics

#TilingTuesday

Tiling from #squares and #triangles

Patterned carpet, Loughborough University, England
#Tiling #TilingTuesday #Pattern #Geometry #MathArt #MathsArt

Tessellations with quarter of octagons and two sizes of squares.
#tilingtuesday #geometry #tiling

Count the #triangles!
View from Pierhead Play Park in North Queensferry, looking toward the Forth Road Bridge and the Queensferry Crossing. #Queensferry #FirthOfForth #Scotland #lines #geometry #BridgesOfMastodon

For those who like gnomons (sundials), a rare book from Cristoforo Clavius published in 1581:

Gnomonices libri octo : in quibus non solum horologiorum solariu[m] sed aliarum quoq[ue] rerum, ...
https://archive.org/details/ARes45105

Watch at: https://youtu.be/jihXKzu8yJo
Play at: https://tic80.com/play?cart=4238

Perfect fit.

Synthesis

Window, North Gate, Imperial Citadel of Thăng Long, Hanoi, Vietnam https://en.wikipedia.org/wiki/Imperial_Citadel_of_Th%C4%83ng_Long#North_Gate_(C%E1%BB%ADa_B%E1%BA%AFc,_or_B%E1%BA%AFc_M%C3%B4n)
#Tiling #TilingTuesday #Pattern #Geometry #MathArt #MathsArt #photography #architecture

Diagonal section of ðe 4d cartesian product of 2 "I want to be a sea urchin and eat cabbage"(yes, really) tilings

kinda like ðis

Unfortunately we are still too dimensionally, perceptually, & probably intellectually challenged to even try to look at ðe full θing

Kepler pentafoil knot #knottheory #math #topology #knots #stem #steam #geometry #mathart #art

It's #TilingTuesday !

Tiling of 12 colourful ducks.

Fun with chiral shapes - cube from snub cubes.

Un pavage de Truchet modifiable en cliquant sur les triangles d'une zone bien précise.
https://www.geogebra.org/m/pngnrsgh