Limb as a spring-loaded pendulum
One effective leg of length L (hip height) and stiffness k carries the body mass. The leg angle at touchdown is found numerically so that the bounce is symmetric.
Spring results
vertical force / weightenergy stored in the leg spring (J/kg)
Diet and tooth wear
remaining crown (%) versus years, all stages, for the chosen diet
Tooth life (years) by stage and diet
Body mass through time (log scale)
stages in this labother genera (approximate)
Toes, tooth height and grassland
front toescrown height / widthgrass cover (illustrative)
Horse evolution is a bush, not a ladder: Anchitherium kept three toes and leafy teeth for 15 million years, Hippidion and Nannippus went their own ways, and body size went down in some lineages. The stages here are snapshots along one successful branch.
Experiment: do gait transitions follow dynamic similarity?
Question: if the six stages (25 to 500 kg) are scaled copies in the dynamical sense, they should change gait at the same Froude number Fr = v²/(g h), which means the transition speed scales as the square root of hip height. The experiment scans speed for each stage and finds where the cheapest gait switches, then compares with the prediction from the reference horse.
The science
What this simulation models
Six snapshots of the horse family: Hyracotherium (Eohippus), Mesohippus, Merychippus, a Hipparion-grade three-toed side branch, a Pliohippus/Dinohippus-grade one-toed horse, and modern Equus. Each snapshot has a Blender-modelled body, foreleg skeleton and cheek tooth; sizes, toe counts and crown heights are rounded literature values. The animation is a procedural gait: foot trajectories are planted on the ground and a two-link inverse-kinematics solver bends the limb, so what you see is consistent with the stride length and duty factor shown in the readouts.
Froude number, stride length and gaits
Fr = v² / (g h) λ / h = 2.3 Fr0.3
h is hip height. Alexander's dynamic-similarity hypothesis: animals of different size move in geometrically similar ways when their Froude numbers match, so relative stride length and gait depend on Fr. The stride law λ/h = 2.3 Fr0.3 (Alexander and Jayes, 1983) gives the stride used for every gait. Duty factors (fraction of the stride a foot is on the ground) are rough linear fits to published ranges and are typical, not exact.
Walking as an inverted pendulum
In walking the body vaults over a stiff leg like an inverted pendulum. With midstance speed v and a stance arc of half-angle α, the vertical force at the end of the arc is m [g (3 cosα - 2) - v²/L]. It reaches zero when
Frmax = 3 cos α - 2
which tends to 1 for short steps and is near 0.5 for the step lengths of a brisk walk. The app integrates the pendulum equation numerically and checks this formula. Real animals switch to trot earlier than the limit, because trot becomes cheaper.
Cost of transport and the gait model
Metabolic power per kilogram P = Prest + Pmech. Walking uses the pendulum collision cost: two redirections of the body per stride, each costing ½ v² sin²α per kg, at 25% muscle efficiency. Trot, canter and gallop use the straight-line power versus speed typical of measurements on horses (Hoyt and Taylor, 1981): each gait has its own intercept and slope, the slope falls as more elastic recoil is used. The lines are written in terms of v/√(g h), so they are dynamically similar; they are calibrated so that a 500 kg horse changes gait at about Fr 0.28 (walk to trot), 2.4 (trot to canter) and 4.1 (canter to gallop), in line with published values for horses. A smaller tendon-storage index raises the slopes slightly (less elastic return). The cheapest gait at each speed is the one the animal uses in automatic mode.
Spring-loaded inverted pendulum and tendon storage
In a bouncing gait the body is a point mass on a spring leg, k = krel m g / L. The app integrates the stance with fourth-order Runge-Kutta, finds the touchdown angle that makes the stance symmetric, and reports contact time, flight time, peak force and the energy stored in the leg spring ½ k (ΔL)². In a horse most of this energy is held by tendons and ligaments in the lower limb (the superficial digital flexor and suspensory ligament), which is why long, light, one-toed limbs with long tendons are efficient. The tendon-storage index per stage (J/kg) is a modelling assumption inferred from limb proportions; tendons do not fossilise.
Toe reduction
The front foot goes from four toes (Hyracotherium) to three (Mesohippus onward) to one (Pliohippus/Dinohippus and Equus). Side toes shrink to splint bones. The Hipparion branch shows that the three-toed foot with short side toes stayed viable for millions of years. The ground-contact view on the Bones tab shows the footprint for each stage.
Teeth, silica and wear
w = κ A (m / 500 kg)0.083 life = 0.85 H / w
Wear depth per year w grows with the abrasive load A (grass silica bodies called phytoliths plus soil grit, in percent of dry food) and only weakly with body size, since food intake scales roughly with m0.75 and tooth area with m2/3. κ is calibrated to about 3 mm per year for a horse on grass. A tall crown (hypsodonty) stores enamel and dentine so the tooth lasts as long as the animal; a leaf-eater's low crown would be worn through in a few years on grass. This is a deliberately simple model: real wear also depends on chewing mechanics, enamel structure and individual food choice. Crown heights and hypsodonty indices are rounded and approximate.
Body mass and the "not a straight line" honesty
The average size of horses did increase from about 25 kg to about 500 kg, but this was not a steady march. The family branched into many genera, some large browsers (Hypohippus), some small three-toed grazers (Nannippus), South American Hippidion, and the main line became larger mostly in the last 20 million years. The grassland curve is an illustrative composite for North American open habitats; C3 grasslands spread through the Miocene and C4 grasses later, and timing differs strongly by region.
What is simplified
- Procedural gait: no muscle forces; the leg-spring is a separate model, not the driver of the animation.
- The gait cost lines are calibrated to a reference horse; other stages inherit them through non-dimensional scaling and a small elastic-storage correction.
- Colours, mane and tail are illustrative; the true coats of fossil horses are not known.
- Stages are snapshots, not an ancestor-descendant chain, and the Pliohippus skeleton is a stand-in for the one-toed grade.
Independent numerical checks built in
1. Inverted-pendulum limit: numerical RK4 integration of the pendulum equation against the closed form 3 cosα - 2. 2. Vertical spring-mass bounce: numerical contact time against the closed-form oscillator solution. 3. Symmetric SLIP stance: energy drift over a stance and the mirror symmetry of the solution. 4. Tooth wear by stepwise time integration against life = usable crown / wear rate. 5. Dynamic similarity of the gait transitions across six body sizes (Experiment tab).
Models and textures: generated in Blender 5.2 (Cycles-baked PBR, Draco GLB). Rendering: Three.js r160. Not a taxonomic reconstruction.