div.stretchy-wrapper {
  width: 100%;
  padding-bottom: 56.25%; /* 16:9 */
  position: relative;
}

  div.stretchy-wrapper > video {
    position: absolute;
    top: 0;
    bottom: 0;
    left: 0;
    right: 0;
    text-align: center;
  }

/* Other aspect ratios to try:
     * 56.25% = 16:9
     * 75% = 4:3
     * 66.66% = 3:2
     * 62.5% = 8:5
     */