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Officine creative Distressed loafers GmxXE
Officine creative Distressed loafers

mathematics and statistics online


Spiral rings #

Traversing the rings one by one in a spiral pattern, we can fill in the interior:

The area of the larger hexagon will be the sum of the circumferences, plus 1 for the center; use this formula to help you calculate the area.

Visiting the hexes this way can also be used to calculate RAGamp;BONE Woman Calla Leather Espadrille Wedge Sandals Light Size 36 Qe5nC

Given a location and a distance, what is visible from that location, not blocked by obstacles? The simplest way to do this is to draw a line to every hex that’s in range. If the line doesn’t hit any walls, then you can see the hex. Mouse over a hex to see the line being drawn to that hex, and which walls it hits.

This algorithm can be slow for large areas but it’s so easy to implement that it’s what I recommend starting with.

There are many different ways to define what’s “visible”. Do you want to be able to see the center of the other hex from the center of the starting hex? Do you want to see any part of the other hex from the center of the starting point? Maybe any part of the other hex from any part of the starting point? Are there obstacles that occupy less than a complete hex? Field of view turns out to be trickier and more varied than it might seem at first. Start with the simplest algorithm, but expect that it may not compute exactly the answer you want for your project. There are even situations where the simple algorithm produces results that are illogical.

Christian Louboutin Womens Studded Clear Pump XhRie
describes a “start at center and move outwards” algorithm to calculate field of view. Also see the Duelo project, which has an an online demo of directional field of view and code on Github. Also see Nike Footscape Flyknit DM Hyper Grape/ o2trI
for an algorithm that works on polygons, including hexagons. For grids, the roguelike community has a nice set of algorithms for square grids (see and and ); some of them might be adapted for hex grids.

For hex to pixel, it’s useful to review the Maison Margiela amp; LaceUp Derbys FuICW
at the top of the page.

For axial coordinates, the way to think about hex to pixel conversion is to look at the . The arrow (0,0)→(1,0) is the q basis vector (x=sqrt(3), y=0) and (0,0)→(0,1) is the r basis vector (x=sqrt(3)/2, y=3/2) . The pixel coordinate is . For example, the hex at (1,1) is the sum of 1 q vector and 1 r vector. A hex at (3,2) would be the sum of 3 q vectors and 2 r vectors.

The code for or is:

This can also be viewed as a matrix multiply, where the basis vectors are the columns of the matrix:

The matrix approach will come in handy later when we want to convert pixel coordinates back to hex coordinates . To invert the process of hex-to-pixel into a pixel-to-hex process, we will invert the hex-to-pixel matrix into a pixel-to-hex matrix.

For offset coordinates, we need to offset either the column or row number (it will no longer be an integer).

Didier Deschamps took charge of his 80th game as France coach – a new record.

Talking point

Argentina coach Jorge Sampaoli’s tactics made little sense. Messi was left isolated in a false-nine role and the ageing defence played too high up the pitch, leaving them exposed to Mbappe’s pace. Aguero’s arrival gave them more penalty-area presence but it was too late.


Possession (%): 41/59

Goal attempts: 9/9

Attempts on target: 4/1

Corners: 0/4

Fouls: 21/15

Pass accuracy (%): 82/85

Distance covered (km): 97/96

Argentina vs Raid Bria Sock Boots RnZkH36hn0
– 26th June Match Report

Match preview here

Argentina Starting XI: 12 Franco Armani, 2 Gabriel Mercado, 3 Nicolas Tagliafico (19 Sergio Aguero 80), 7 Ever Banega, 9 Gonzalo Higuain, 10 Lionel Messi, 11 Angel Di Maria (13 Max Meza 72), 14 Javier Mascherano, 15 Enzo Perez (22 Cristian Pavon 61), 16 Marcos Rojo, 17 Nicolas Otamendi

Nigeria Starting XI: 23 Francis Uzoho, 2 Brian Idowu, 4 Wilfred Ndidi, 5 William Troost-Ekong, 6 Leon Balogun, 7 Ahmed Musa (13 Simeon Nwankwo 90+2), 8 Ohenekaro Etebo, 10 Jon Obi Mikel, 11 Victor Moses, 14 Kalechi Iheanacho (9 Odion Ighalo 46), 22 Kenneth Omeruo (18 Alex Iwobi 90)

St Petersburg

Argentina 2 Nigeria 1

Argentina: Messi 14,Rojo 86

Nigeria: Moses 51pen

Macos Rojo was the unexpected hero who kept Argentina in the tournament with a brilliant late volley to beat Nigeria and claim second place in the group behind Croatia. It was a devastating blow for Nigeria, who had been more than a match for Argentina.

A refocused Lionel Messi gave Argentina the lead with an outstanding first-half goal but defensive mistakes allowed Nigeria back in the game through a Victor Moses penalty.

As the clock ticked down to an Argentinian exit, Gonzalo Higuain put an easy chance over the bar, while Odion Ighalo had two opportunities to put Nigeria beyond reach before Rojo’s sensational strike.

Lionel Messi scored a sublime goal against Nigeria (Getty Images)

Never in doubt. Was it? #NGAARG Badgley Mischka Womens Hampden Crystal Embellished Sandal 4J3kraBusE
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— FIFA World Cup (@FIFAWorldCup) Office OSLO Mules blue/turquoise QLYG3bWi

Lionel Messi became the first player to score a World Cup goal as a teenager, in his twenties and in his thirties.


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A budget car first and foremost

Photo credit: Josiah Bondy

The 2018 Nissan Leaf is an all-electric that has been around the block a few times.

Not as many times as you might think, since it has much lower range (at 151 miles) than the Chevy Bolt (at 238 miles) and the fully equipped Tesla Model S (at 315 miles).

Yet, in a recent test, there are a few high-tech options that will make you wonder if this might be the right EV for you.

Photo credit: Josiah Bondy

First, you should know that the Leaf is a budget car in every sense of the word. It’s meant to be light, not that sporty, small, and a bit short on tech features.

The amazing touchscreen display on the Bolt and Model S help you manage all of the settings and see what's happening with the electric motor at all times. The Leaf, not so much.

The display is small and a little disappointing – you won’t see any animations of a car that shows how regenerative braking works.

Instead, you’ll see a bland blue and black display that shows simple data feeds like the range and how many extra miles you can gain from turning off the climate control settings.

Photo credit: Josiah Bondy

For any tech connoisseur, it might be a letdown given the trend in cars to provide richly detailed touchscreens, both above the steering wheel and in the dash.

The Leaf debuted in 2010, and the new model has a few more curves and enhancements, but the displays look roughly the same.

Fortunately, there’s a few other features to keep you occupied. One is a new E-Pedal that allows you to drive the car without using the brakes.

Essentially, once activated, you can drive the car by pushing the accelerator to speed up, or laying off the accelerator to slow down to a complete stop. It worked like second nature after a little practice.

It’s similar to a feature you’ll find on some all-terrain vehicles like those made by Polaris called Engine Braking.

“The Leaf’s E-Pedal, which comes standard across all grade levels, allows the driver to accelerate, decelerate and stop by the car using the accelerator pedal alone,” says Jeff Wandell, a Nissan Leaf spokesperson. “The Leaf comes to a smooth and complete stop by simply releasing the accelerator. Furthermore, E-Pedal will even hold the vehicle in place on an incline – a feature only found on the Leaf.”

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Alignments to Content Standards: G-CO.A.1
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Three students have proposed these ways to describe when two lines and are perpendicular:

Explain why each of these definitions is correct. What are some of the advantages and disadvantages with each?

IM Commentary

The purpose of this task is to critically examine some different possible definitions of what it means for two lines to be perpendicular. All of these definitions are correct but they convey different information and build on different knowledge. Definition (a) is the simplest and most common and it is very closely related to (b). Though very similar, Definition (a) works from a weaker hypothesis than Definition (b) because the fact that the four angles made by perpendicular lines are right can be deduced, using supplementary and/or vertical angles, from the fact that of angles is right. Since definitions should be as simple as possible, the weaker hypothesis of Definition (a) is preferable.

Definition (c) is very different in flavor from the first two. While it is mathematically correct, it is not the best definition because defining the reflection about requires knowing what it means for two lines to be perpendicular. Unless reflection about can be defined in some other way which does not use perpendicular lines, this leads to circular reasoning. Hence, although it is important to understand (c), this would not be a good choice for defining perpendicular lines. Nonetheless, reflections are fundamentally related to the notion of perpendicular lines and so it is very appropriate to consider the two together, understanding that perpendicular lines are a logical prerequisite to define reflections.

Work on finding the clearest, most economical definitions embodies MP6, Attend to Precision. Mathematical definitions need to be unambiguous and, ideally, they should convey the necessary information without unneeded hypotheses. Work on this task is ideally suited for class discussion as different students will likely react differently to the definitions and, through discussion, they will be able to come to a deeper understanding of what it means for lines to be perpendicular.


Below is a picture of this situation which we will use for (a) and then (b). Here we have assumed that is a right angle.

The main advantage to this definition is its simplicity. It only makes an assumption about one of the four angles at , as opposed to Definition (b) which assumes that all four are right angles. Also, unlike Definition (c) the only prior knowledge assumed is the definition of a right angle.

We also show here that Definition (a) implies Definition (b) (with fewer hypotheses). Note that angles and are supplementary since together they form . Since by hypothesis and this means that , so it is also a right angle. The same reasoning can be applied to conclude that since, together with , this makes line . Finally, for , it makes a vertical pair together with right angle and so it is congruent to . Therefore . Thus all four angles made by and at are right angles.


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