In a medium bowl combine flour, baking powder and salt, set aside.
In a large bowl, beat sugar and butter with an electric mixer, until light and fluffy. Add eggs one at a time, beating after each addition, then add vanilla. Gradually add the flour mixture, beating on low speed, just until incorporated.
Divide dough into 3 equal portions. Shape one portion into a 4 x 4-inch square, wrap in plastic wrap and place in the refrigerator, this will be the white portion of the cookie.
Return one portion to the bowl and add red food coloring, adding more to reach desired color. Shape into a 4 x 4-inch square, wrap in plastic wrap and place in the refrigerator. Repeat steps with the last portion of dough, adding blue food coloring, to reach desired color. Refrigerate all of pieces of dough for 1 hour.
Remove the dough from the refrigerator. Cut each square in half to make two rectangles. Wrap half of each color and place back in the refrigerator. Roll out white dough between two pieces of wax paper until the rectangle is about 1/8-inch thick. Repeat with red and blue rectangles.
Peel the wax paper off one side of the red dough. Peel the wax paper off one side of the white dough. Lay the white dough on top of the red dough and peel back the wax paper off the white dough. Peel the wax paper off one side of the blue dough and lay on top of the white dough. Remove the wax paper from the other side of the blue dough.
Starting at the short end, roll all three layers tightly together to form a log. Repeat the stacking and rolling process with the remaining dough in the refrigerator. Wrap the logs in wax paper and then in plastic wrap, refrigerate for 1 hour.
Remove the logs from the refrigerator and roll them on the counter several times to remove the flat side. Refrigerate the logs for an additional 1-4 hours.
When ready to bake, preheat the oven to 350 degrees, and line 2 baking sheets with parchment paper. Remove the logs one at a time and slice into 1/4-inch rounds. Place the rounds about 2-inches apart on the baking sheet. Bake for 9-11 minutes, until puffed and lightly golden (not brown). Cool on baking sheets for 5 minutes then transfer to a cooling rack to cool completely. Store the pinwheels in an airtight container.
Cook's Note: The dough will be sticky, do not add more flour as the cookies will be tough. Refrigerating the dough makes it easier to shape and cut the cookies.
To make these cookies even more festive, roll the logs in sprinkles before cutting into cookies and bake as directed above.