Two-Dimensionality as an Ontological Filter: Why △₁ₓ₁ Is Not an Arbitrary Choice

This work develops the key idea of △-ontology: the fundamental object of mathematics is the right isosceles triangle △₁ₓ₁, not for aesthetic or convenient reasons, but because two-dimensionality is an ontological filter through which reality enters consciousness. The key logic of the work. Two-dimensionality of perception explains the dimensionality of the base object, but does not explain its shape. The shape △₁ₓ₁ (right isosceles) is explained separately — by the principle of energy optimality: △₁ₓ₁ is the only figure that simultaneously (a) has a right angle (orthogonality), (b) has equal legs (symmetry), (c) has an irrational hypotenuse √2 (connection with the continuum), and (d) has self-similarity via the altitude from the right angle (invertible Φ ∘ Ψ = id). Two-dimensionality chooses the dimensionality; energy optimality chooses the shape. We show that: 1. All spatial receptor surfaces are topologically two-dimensional (retina, skin, cochlea). 2. Pressure is the primary notion relative to mass, and it is described by the formula p = F / S, where S is the area of a flat membrane. 3. Light is a transverse plane wave, which makes two-dimensionality physically fundamental. 4. Chemical senses (smell, taste) work differently — they transmit not spatial information but valent information (properties of molecules), and therefore do not cancel but complement the two-dimensional filter. 5. △₁ₓ₁ is the only object possessing invertible self-similarity Φ ∘ Ψ = id, which makes it the natural attractor for a two-dimensional cognitive system. 6. Two-dimensionality is the arena of provability: only in 2D is complete invertibility of cutting and gluing possible. Conclusion. △₁ₓ₁ is not an arbitrary choice but a necessary consequence of two independent principles: (1) two-dimensionality of spatial perception (determines dimensionality) and (2) invertible self-similarity Φ ∘ Ψ = id (determines shape). Chemical senses (smell, taste) work on a different — valent — level and do not cancel but complement △-ontology.