Naming angles-of-polygons
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This document provides information about polygons, including definitions of key terms like vertex, diagonal, and consecutive sides and vertices. It explains that polygons are named based on the number of sides, with examples given for triangles, quadrilaterals, pentagons, etc. up to dodecagons. Formulas are provided relating the number of sides of a polygon to the sum of its interior angles and the measure of one interior angle. The terms exterior and interior angles are defined, and an example shows how to use the formula that the sum of exterior angles is 360 degrees to find the measure of one exterior angle of a regular hexagon.
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The Milky Way’s (MW) inner stellar halo contains an [Fe/H]-rich component with highly eccentric orbits, often referred to as the
‘last major merger.’ Hypotheses for the origin of this component include Gaia-Sausage/Enceladus (GSE), where the progenitor
collided with the MW proto-disc 8–11 Gyr ago, and the Virgo Radial Merger (VRM), where the progenitor collided with the
MW disc within the last 3 Gyr. These two scenarios make different predictions about observable structure in local phase space,
because the morphology of debris depends on how long it has had to phase mix. The recently identified phase-space folds in Gaia
DR3 have positive caustic velocities, making them fundamentally different than the phase-mixed chevrons found in simulations
at late times. Roughly 20 per cent of the stars in the prograde local stellar halo are associated with the observed caustics. Based
on a simple phase-mixing model, the observed number of caustics are consistent with a merger that occurred 1–2 Gyr ago.
We also compare the observed phase-space distribution to FIRE-2 Latte simulations of GSE-like mergers, using a quantitative
measurement of phase mixing (2D causticality). The observed local phase-space distribution best matches the simulated data
1–2 Gyr after collision, and certainly not later than 3 Gyr. This is further evidence that the progenitor of the ‘last major merger’
did not collide with the MW proto-disc at early times, as is thought for the GSE, but instead collided with the MW disc within
the last few Gyr, consistent with the body of work surrounding the VRM.
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Or: Beyond linear.
Abstract: Equivariant neural networks are neural networks that incorporate symmetries. The nonlinear activation functions in these networks result in interesting nonlinear equivariant maps between simple representations, and motivate the key player of this talk: piecewise linear representation theory.
Disclaimer: No one is perfect, so please mind that there might be mistakes and typos.
dtubbenhauer@gmail.com
Corrected slides: dtubbenhauer.com/talks.html
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Naming angles-of-polygons
- 3. What ddooeess tthhee wwoorrdd ““ppoollyyggoonn”” mmeeaann?? WWhhaatt iiss tthhee ssmmaalllleesstt nnuummbbeerr ooff ssiiddeess aa ppoollyyggoonn ccaann hhaavvee?? WWhhaatt iiss tthhee llaarrggeesstt nnuummbbeerr ooff ssiiddeess aa ppoollyyggoonn ccaann hhaavvee??
- 4. Triangle Quadrilateral Pentagon Hexagon Heptagon Octagon Nonagon Decagon Dodecagon n-gon
- 5. Hip Bone’s connected to the… Classifying Polygons Polygons with 3 sides… Triangles Polygons with 4 sides… Quadrilaterals Polygons with 5 sides.. Pentagons But wait we have more polygons Polygons with 6 sides… Hexagons Polygons with 7 sides… Heptagons Polygons with 8 sides… Octagons But still we have more polygons Polygons with 9 sides… Nonagons Polygons with 10 sides… Decagons Polygons with 12 sides… Dodecagons And now we have our polygons
- 6. F Important Terms A B C E D A VERTEX is the point of intersection of two sides A segment whose endpoints are two nonconsecutive vertices is called a DIAGONAL. CONSECUTIVE VERTICES are two endpoints of any side. Sides that share a vertex are called CONSECUTIVE SIDES.
- 7. More Important Terms EQUILATERAL - All sides are congruent EQUIANGULAR - All angles are congruent REGULAR - All sides and angles are congruent
- 8. Polygons are named by listing its vertices consecutively. A B F C E D
- 9. # of sides # of triangles Sum of measures of interior angles 3 1 1(180) = 180 4 2 2(180) = 360 5 3 3(180) = 540 6 4 4(180) = 720 n n-2 (n-2) ·180
- 10. If a convex polygon has n sides, then the sum of the measure of the interior angles is (n – 2)(180°)
- 11. Ex. 1 Use the regular pentagon to answer the questions. A)Find the sum of the measures of the interior angles. 540° B)Find the measure of ONE interior angle 108°
- 12. Two more important terms Exterior Angles Interior Angles
- 13. If any convex polygon, the sum of the measures of the exterior angles, one at each vertex, is 360°. 1 2 3 4 5 mÐ1+mÐ2 +mÐ3+mÐ4 +mÐ5 = 360
- 14. If any convex polygon, the sum of the measures of the exterior angles, one at each vertex, is 360°. 1 3 2 mÐ1+mÐ2 +mÐ3 = 360
- 15. If any convex polygon, the sum of the measures of the exterior angles, one at each vertex, is 360°. 1 3 2 4 mÐ1+mÐ2 +mÐ3+mÐ4 = 360
- 16. Ex. 2 Find the measure of ONE exterior angle of a regular hexagon. sum of the exterior angles number of sides = 60° = 360 6
- 17. Ex. 4 Each exterior angle of a polygon is 18°. How many sides does it have? exterior angle sum of the exterior angles = n = 20 number of sides 360 =18 n