1ucasvb:

The familiar trigonometric functions can be geometrically derived from a circle.
But what if, instead of the circle, we used a regular polygon?
In this animation, we see what the “polygonal sine” looks like for the square and the hexagon. The polygon is such that the inscribed circle has radius 1.
We’ll keep using the angle from the x-axis as the function’s input, instead of the distance along the shape’s boundary. (These are only the same value in the case of a unit circle!) This is why the square does not trace a straight diagonal line, as you might expect, but a segment of the tangent function. In other words, the speed of the dot around the polygon is not constant anymore, but the angle the dot makes changes at a constant rate.
Since these polygons are not perfectly symmetrical like the circle, the function will depend on the orientation of the polygon.
More on this subject and derivations of the functions can be found in this other post
Now you can also listen to what these waves sound like.
This technique is general for any polar curve. Here’s a heart’s sine function, for instance

1ucasvb:

The familiar trigonometric functions can be geometrically derived from a circle.

But what if, instead of the circle, we used a regular polygon?

In this animation, we see what the “polygonal sine” looks like for the square and the hexagon. The polygon is such that the inscribed circle has radius 1.

We’ll keep using the angle from the x-axis as the function’s input, instead of the distance along the shape’s boundary. (These are only the same value in the case of a unit circle!) This is why the square does not trace a straight diagonal line, as you might expect, but a segment of the tangent function. In other words, the speed of the dot around the polygon is not constant anymore, but the angle the dot makes changes at a constant rate.

Since these polygons are not perfectly symmetrical like the circle, the function will depend on the orientation of the polygon.

More on this subject and derivations of the functions can be found in this other post

Now you can also listen to what these waves sound like.

This technique is general for any polar curve. Here’s a heart’s sine function, for instance

(via diy)

modcloth:

Brit + Co’s 25 watermelon recipes are the perfect summer go-to. Watermelon is such an amazing summer treat!
<3 Amy, ModStylist
modcloth:

Brit + Co’s 25 watermelon recipes are the perfect summer go-to. Watermelon is such an amazing summer treat!
<3 Amy, ModStylist

modcloth:

Brit + Co’s 25 watermelon recipes are the perfect summer go-to. Watermelon is such an amazing summer treat!

<3 Amy, ModStylist



Two dogs run into the field in a Galatasaray friendly match against Ahlen.



Two dogs run into the field in a Galatasaray friendly match against Ahlen.



Two dogs run into the field in a Galatasaray friendly match against Ahlen.



Two dogs run into the field in a Galatasaray friendly match against Ahlen.



Two dogs run into the field in a Galatasaray friendly match against Ahlen.



Two dogs run into the field in a Galatasaray friendly match against Ahlen.



Two dogs run into the field in a Galatasaray friendly match against Ahlen.



Two dogs run into the field in a Galatasaray friendly match against Ahlen.

Two dogs run into the field in a Galatasaray friendly match against Ahlen.

(via collegehumor)

zodiaccity:

Rest In Peace Talia! &lt;3 1999-2013

zodiaccity:

Rest In Peace Talia! <3 1999-2013

(via cakedup-makeup)

instagram:


Swimming with Swine on “Pig Island&#8221;
For more photos from “Pig Island,&#8221; check out the #swimmingpigs hashtag. 
The nation of the Bahamas consists of more than 700 islands. While the tropical climate and pristine beaches are invitation enough to make the Bahamas a prime tourist destination, one island has a particular draw: swimming pigs.
Big Major Cay is a small island that is uninhabited with the exception of a few dozen pigs. Though quite popular, no one quite seems to know how they got there in the first place. Local lore, however, suggests they may have been the sole survivors of a nearby shipwreck.
As visitors come and bring snacks for the animals, the pigs have come to view visiting tourists as a dependable food source. As a result, the pigs will actually swim out to oncoming boats to greet them as they approach in hopes of a handout.
instagram:


Swimming with Swine on “Pig Island&#8221;
For more photos from “Pig Island,&#8221; check out the #swimmingpigs hashtag. 
The nation of the Bahamas consists of more than 700 islands. While the tropical climate and pristine beaches are invitation enough to make the Bahamas a prime tourist destination, one island has a particular draw: swimming pigs.
Big Major Cay is a small island that is uninhabited with the exception of a few dozen pigs. Though quite popular, no one quite seems to know how they got there in the first place. Local lore, however, suggests they may have been the sole survivors of a nearby shipwreck.
As visitors come and bring snacks for the animals, the pigs have come to view visiting tourists as a dependable food source. As a result, the pigs will actually swim out to oncoming boats to greet them as they approach in hopes of a handout.
instagram:


Swimming with Swine on “Pig Island&#8221;
For more photos from “Pig Island,&#8221; check out the #swimmingpigs hashtag. 
The nation of the Bahamas consists of more than 700 islands. While the tropical climate and pristine beaches are invitation enough to make the Bahamas a prime tourist destination, one island has a particular draw: swimming pigs.
Big Major Cay is a small island that is uninhabited with the exception of a few dozen pigs. Though quite popular, no one quite seems to know how they got there in the first place. Local lore, however, suggests they may have been the sole survivors of a nearby shipwreck.
As visitors come and bring snacks for the animals, the pigs have come to view visiting tourists as a dependable food source. As a result, the pigs will actually swim out to oncoming boats to greet them as they approach in hopes of a handout.
instagram:


Swimming with Swine on “Pig Island&#8221;
For more photos from “Pig Island,&#8221; check out the #swimmingpigs hashtag. 
The nation of the Bahamas consists of more than 700 islands. While the tropical climate and pristine beaches are invitation enough to make the Bahamas a prime tourist destination, one island has a particular draw: swimming pigs.
Big Major Cay is a small island that is uninhabited with the exception of a few dozen pigs. Though quite popular, no one quite seems to know how they got there in the first place. Local lore, however, suggests they may have been the sole survivors of a nearby shipwreck.
As visitors come and bring snacks for the animals, the pigs have come to view visiting tourists as a dependable food source. As a result, the pigs will actually swim out to oncoming boats to greet them as they approach in hopes of a handout.
instagram:


Swimming with Swine on “Pig Island&#8221;
For more photos from “Pig Island,&#8221; check out the #swimmingpigs hashtag. 
The nation of the Bahamas consists of more than 700 islands. While the tropical climate and pristine beaches are invitation enough to make the Bahamas a prime tourist destination, one island has a particular draw: swimming pigs.
Big Major Cay is a small island that is uninhabited with the exception of a few dozen pigs. Though quite popular, no one quite seems to know how they got there in the first place. Local lore, however, suggests they may have been the sole survivors of a nearby shipwreck.
As visitors come and bring snacks for the animals, the pigs have come to view visiting tourists as a dependable food source. As a result, the pigs will actually swim out to oncoming boats to greet them as they approach in hopes of a handout.
instagram:


Swimming with Swine on “Pig Island&#8221;
For more photos from “Pig Island,&#8221; check out the #swimmingpigs hashtag. 
The nation of the Bahamas consists of more than 700 islands. While the tropical climate and pristine beaches are invitation enough to make the Bahamas a prime tourist destination, one island has a particular draw: swimming pigs.
Big Major Cay is a small island that is uninhabited with the exception of a few dozen pigs. Though quite popular, no one quite seems to know how they got there in the first place. Local lore, however, suggests they may have been the sole survivors of a nearby shipwreck.
As visitors come and bring snacks for the animals, the pigs have come to view visiting tourists as a dependable food source. As a result, the pigs will actually swim out to oncoming boats to greet them as they approach in hopes of a handout.
instagram:


Swimming with Swine on “Pig Island&#8221;
For more photos from “Pig Island,&#8221; check out the #swimmingpigs hashtag. 
The nation of the Bahamas consists of more than 700 islands. While the tropical climate and pristine beaches are invitation enough to make the Bahamas a prime tourist destination, one island has a particular draw: swimming pigs.
Big Major Cay is a small island that is uninhabited with the exception of a few dozen pigs. Though quite popular, no one quite seems to know how they got there in the first place. Local lore, however, suggests they may have been the sole survivors of a nearby shipwreck.
As visitors come and bring snacks for the animals, the pigs have come to view visiting tourists as a dependable food source. As a result, the pigs will actually swim out to oncoming boats to greet them as they approach in hopes of a handout.

instagram:

Swimming with Swine on “Pig Island”

For more photos from “Pig Island,” check out the #swimmingpigs hashtag.

The nation of the Bahamas consists of more than 700 islands. While the tropical climate and pristine beaches are invitation enough to make the Bahamas a prime tourist destination, one island has a particular draw: swimming pigs.

Big Major Cay is a small island that is uninhabited with the exception of a few dozen pigs. Though quite popular, no one quite seems to know how they got there in the first place. Local lore, however, suggests they may have been the sole survivors of a nearby shipwreck.

As visitors come and bring snacks for the animals, the pigs have come to view visiting tourists as a dependable food source. As a result, the pigs will actually swim out to oncoming boats to greet them as they approach in hopes of a handout.