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Geometric Curved Checkerboard
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This Photoshop tutorial might take you back to a math class you had in school. The problem was how to create curves using only straight lines. Here is how I did it.

First let's create a new file 350x350px. Choose View/Show/Grid. Also make sure to checkmark View/Snap. You should have something that looks like this.

The next thing you wll want to do is to create a border. We will want to create this on a separate layer. Select Layer/New/Layer and click OK. Select the pencil tool (shortcut key B) and set the master diameter to 2 pixels. While holding the shift key down to keep the line straight, follow along one square inside the grid to create the border. Click and drag each line as you go.

You should end up with this. Click Ctrl+' to turn the grid off and on.

Turn the grid back on as we will need it to complete this project. Select the line tool with a weight of 1px.

From now on it is just a simple pattern of lines. I will show the first few so you get the idea. Using our border, drag the first line from the bottom left and drag to the second point in the upper left.

The next line will be from the 2nd point from the bottom left to the 3rd point on the upper left.

And one more sample just to get the pattern embedded into your head. This line will be from the 3rd point up from the bottom left to the 4th point on the upper left.

Continue the pattern all the way up until you get this.

Turn the grid off to see what we have so far. This looks pretty slick to me.

Now let's repeat this process to create the same geometrical curve on the bottom right side of the document. Here is what I came up with.

Now we need to add a little color to the mix. I am going to use the magic wand tool (shorcut key W) to select a checkerboard pattern to fill with red. Make sure you have the "add to selection" box highlighted.

Here is a picture of a few of the boxes selected. continue on in this pattern until they are all selected.

Make sure that your foreground tile is whichever color you want your checkerboard to be. I have chosen red.

Now choose Edit/Fill and select Foreground Color from the drop down menu and you should get this.

OK now that is looking pretty sweet. I think just one more little spice and this dish will be done. Select Layer/Layer Style/Blending Options and check the Bevel and Emboss box. The default settings should be just fine for this.

Now this is math meets art. Do you think someone would actually say that there are no curved lines here? Well you now know the answer. Here is a picture that I created using these same techniques.