The classic Minkowski hyperbolic framework is often hard to grasp for those who get involved in relativity theory. Euclidean relativity presents an intuitive circular geometry as alternative. It uses proper time t as the fourth spatial dimension and shows that all objects in 4D have universal velocity c.

The links section gives a brief overview of its characteristics and a number of references to articles on Euclidean relativity published by various authors. Each author approaches the topic in his own way and individual interpretations were often developed independently without knowledge of the other authors. First fundamental work came from the Dutch mathematician Hans Montanus in the beginning of the 90's. During the last 5-10 years other authors have begun to find each other and have published their articles, initiating an innovative trend in relativity.

Below you will also find my own interpretation of Euclidean special relativity. It provides arguments for a geometrical unification of gravity and electromagnetism in five dimensions, which is worked out in another article. From this unification follows the proposal to regard mass particles themselves, instead of gravitons, as bosons following null geodesics in 5D gravity.

The full text of these articles is available in web form and as downloadable PDF documents. A simplified and popularized description of the essentials of Euclidean relativity is also available.

One of the articles describes a fractal-like universe, based on the geometry of Euclidean relativity. It suggests a hierarchical ordering of the four forces of Nature together with their fermions and bosons through their number of dimensions.
 

Articles   (see the Links section for articles of other authors)

Dimensions in Special Relativity Theory- A Euclidean Interpretation
A Euclidean interpretation of special relativity is given wherein proper time t  acts as the fourth Euclidean coordinate and time t becomes a fifth Euclidean dimension. Velocity components in both space and time are formalized while their vector sum in four dimensions has invariant magnitude c. Classical equations are derived from this Euclidean concept. The velocity addition formula shows a deviation from the standard one; an analysis and justification is given for that.
Full text: in PDF (September 2005) or online in web form  (Galilean Electrodynamics Vol 18 nr 1, Jan/Feb 2007).

Mass Particles as Bosons in Five Dimensional Euclidean Gravity
Velocities in both space and time are formalized and generalized for n-dimensional Euclidean spaces. Applied to photons and mass particles, this implies a dimensional hierarchy between their velocities. It is suggested that mass particles, that are fermions in 4D, behave like bosons in 5D gravity where they follow null geodesics.
Full text: in PDF (Revised May 2007) or online in web form.

Minkowski versus Euclidean 4-vectors
Minkowski 4-vectors are written in Euclidean form. It is shown that in this way the physical meaning of their components can be made more intuitive and directly associated with geometric properties in Euclidean space-time.
Full text: in PDF (February 2006) or online in web form.

Propulsion without propellant using four-momentum of photons in Euclidean special relativity
An alternative method to accelerate particles or objects is described. It uses principles of 4D momentum that follow from Euclidean special relativity.
Full text: in PDF (April 2008) or online in web form.

Relativity Simplified
Relativity has always been taught using Minkowski's geometry, where the time dimension is markedly distinguished from spatial dimensions. It is possible though, to describe the effects of relativity using a pure Euclidean geometry where time and spatial dimensions are essentially identical in nature. This is a laymen's version of such a description that provides simple and intuitive explanations for most relativistic phenomena.
First, the concept of speed in 2, 3 and 4 dimensions is explained, meanwhile introducing the notion of speed in the time dimension. This is then used to show the mechanisms behind relativistic time dilation and length contraction. The last sections show why the speed of light remains constant in all situations, followed by a brief overview of conclusions from the main articles on Euclidean relativity.
Full text: in PDF (Updated March 2008) or online in web form.

The Universe as a Multi-Dimensional Fractal
From various points of view a fractal-like universe is described. Unlike in usual fractals, recurring patterns correlate with the number of dimensions in the observation, i.e., zooming out and in occurs by adding or removing dimensions rather than changing scale. In this way, the forces of Nature together with their fermions and bosons are ordered hierarchically through their number of dimensions. The first three sections show some examples of physical phenomena that re-appear in spaces with one dimension less or more. From section four onwards, they will be put in a broader context, covering all of Nature's forces and particles.
Full text: in PDF (Updated October 2010) or online in web form.

3D graph for relativistic addition of velocities as derived from the Euclidean interpretation

 
3D graph of classical equation for relativistic addition of velocities