This page uses beta as speed divided by the speed of light. The Lorentz factor is one divided by the square root of one minus beta squared. Coordinate length in the gate rest frame is rod rest length divided by that factor. Begin at beta zero, where gamma is one. The rod rest length is one hundred twenty meters, and its gate frame length is also one hundred twenty meters. The two bars initially have equal width, four hundred eighty pixels, using four pixels per meter. Increase beta from zero to point eight. The top marker tracks speed fraction. The cyan rest length stays fixed, while the yellow gate frame coordinate length follows the exact Lorentz formula. It contracts continuously because simultaneous endpoint measurements in this frame are different from simultaneous measurements in the rod frame. This is a coordinate comparison, not a photographic image, a material squeeze or a retarded light ray rendering. Hold beta at point eight. Gamma is five thirds, and the gate frame rod length is seventy two meters. Its yellow bar is now two hundred eighty eight pixels wide. Rest length stays one hundred twenty meters. The gate is one hundred meters in its own rest frame, so the contracted rod fits in this simultaneous coordinate comparison. In the probe rest frame, the rod keeps its rest length and the gate contracts instead. The two frames use different simultaneity conditions; neither mechanically crushes the rod. The same Lorentz factor can be derived using an explanatory light clock triangle. Choose a vertical proper light path of three units and a shared scale of seventy pixels per unit. Begin at beta zero. The horizontal displacement is zero, the vertical path is three units and the diagonal path is also three units. Lab time divided by clock proper time is one. This constructed triangle explains the formula; it is not a calibrated measurement of the photon animation on this page. Increase beta from zero to point eight. The vertical proper path stays three units. The diagonal light path grows to gamma times three, and the horizontal clock displacement is beta times that diagonal length. These instantaneous dimensions satisfy the right triangle relation. The moving corner follows the formulas using the actual current beta; the picture is not simply revealing a finished three four five triangle. Hold beta at point eight. The vertical, horizontal and diagonal lengths are three, four and five units. Their drawn lengths are two hundred ten, two hundred eighty and three hundred fifty pixels. Three squared plus four squared equals five squared. The horizontal to diagonal ratio gives beta, four fifths, and the diagonal to vertical ratio gives gamma, five thirds. The corner remains still while these numbers are spoken. The source animation has chosen coordinates and speeds, so its visual photon motion should not be treated as independently calibrated physical evidence. The source approaching longitudinal Doppler factor is the square root of one plus beta divided by one minus beta. It means observed frequency divided by emitted frequency for the stated approaching geometry. Start at beta zero: the factor is one, so there is no frequency shift. The upper marker shows speed fraction and the lower marker shows the calculated ratio on a shared formula curve. The ratio axis has zero at its baseline and three at the top. Increase beta from zero to point eight. The calculated approaching frequency ratio rises along the curve. Each marker position is computed from the actual changing speed fraction. This assumes longitudinal approach. A receding source would instead use the inverse ratio, so direction is part of the physical statement, not a detail to omit. The page star colors are illustrative categories, not computed spectra. Hold beta at point eight. The approaching ratio is three: observed frequency is three times emitted frequency under that stated condition. The inverse receding ratio would be one third. The markers remain fixed during the numerical explanation. The page Terrell sphere is a constructed rotated and contracted mesh, not a full differential light travel time solver. Its JSON always exports the rod gate frame contracted length, even with probe frame selected. Reset preserves speed, frame and toggles while clearing animation progress. Original controls, camera gestures and snapshot export remain available above this video.