Calculation of Elapsed Time From The Doppler Diagram
The twin on the ship sees low frequency (red) images for 3 years. During that time, he would see the Earth twin in the image grow older by 3/3 = 1 years. He then sees high frequency (blue) images during the back trip of 3 years. During that time, he would see the Earth twin in the image grow older by 3 × 3 = 9 years. When the journey is finished, the image of the Earth twin has aged by 1 + 9 = 10 years.
The Earth twin sees 9 years of slow (red) images of the ship twin, during which the ship twin ages (in the image) by 9/3 = 3 years. He then sees fast (blue) images for the remaining 1 year until the ship returns. In the fast images, the ship twin ages by 1 × 3 = 3 years. The total aging of the ship twin in the images received by Earth is 3 + 3 = 6 years, so the ship twin returns younger (6 years as opposed to 10 years on Earth).
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