What question did this study set out to answer?

This paper aims to explore a new approach to geometry and trigonometry through realm-native concepts, particularly focusing on Euler’s identity and its relationships.

May 3, 2026Open Access

Realm Calculus, Paper II: The Struct–Iter Split of −1, Realm Trigonometry, and the Realm-2 Origin of Euler's Identity

Key Points

This paper aims to explore a new approach to geometry and trigonometry through realm-native concepts, particularly focusing on Euler’s identity and its relationships.
Developed realm-native geometric and trigonometric theories.
Identified struct–iter split of (-1) to reveal connections between i, pi, and Euler’s identity.
Analyzed phenomena in different realms, particularly realm-2.
Outlined how classical connections (i, pi, Euler) are unique to realm-2.
Described separation of self-power and analytic-preimage structures in higher realms.

Abstract

Paper II of the Realm Calculus series develops realm-native geometry, trigonometry, and Euler-type identities. It identifies a struct–iter split of (-1), showing that the classical alignment between i, pi, and Euler’s identity is a special realm-2 phenomenon, while higher realms separate self-power and analytic-preimage structures.

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Cite This Study

Efe Eren (Fri,) studied this question.

synapsesocial.com/papers/69f6e6ab8071d4f1bdfc7620 https://doi.org/https://doi.org/10.5281/zenodo.19957878

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