Related papers: Moon in a puddle and the four-vertex theorem
In this paper, we provide an easy proof of the Four-colour Theorem in a special case indeed.
We prove that a simple closed plane curve with curvature at most 1 and diameter at least 4, surrounds two disjoint open unit disks. We also pose an open question relating the diameter lower bound to the length of the curve.
We give a pen and paper and (comparatively) much simpler proof to verify of the Four Colour Theorem.
In this note we fill a gap in the proof of the main theorem (Theorem 1.2) of our paper 'Surfaces in 4-manifolds', Math. Res. Letters 4 (1997), 907-914.
We give a geometric proof of the Routh's theorem for tetrahedra.
We investigate vertices for plane curves with singular points. As plane curves with singular points, we consider Legendre curves (respectively, Legendre immersions) in the unit tangent bundle over the Euclidean plane and frontals…
This paper provides a new simple proof of Hesse's theorem in projective geometry for any dimension.
A multivariate Gauss-Lucas theorem is proved, sharpening and generalizing previous results on this topic. The theorem is stated in terms of a seemingly new notion of convexity. Applications to multivariate stable polynomials are given.
We give a geometric approach to the proof of the $\lambda$-lemma. In particular, we point out the role pseudoconvexity plays in the proof.
We make an attempt at proving the Four Colour Theorem in six pages.
The Moon often appears larger near the perceptual horizon and smaller high in the sky though the visual angle subtended is invariant. We show how this illusion results from the optimization of a projective geometrical frame for conscious…
In this paper, we show the new fixed point theorem in metric spaces. Furthermore, for this fixed point theorem, we apply to the Collatz conjecture.
In this note, we give an alternate proof of the multinomial theorem using a probabilistic approach. Although the multinomial theorem is basically a combinatorial result, our proof may be simpler for a student familiar with only basic…
Here we outline a proof for the 4-dimensional smooth Poincare Conjecture.
We prove the Yoneda lemma inside an elementary higher topos, generalizing the Yonda lemma for spaces.
This paper presents a short and simple proof of the Four-Color Theorem that can be utterly checkable by human mathematicians, without computer assistance. The new key idea that has allowed it and the global structure of the proof are…
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There are many four vertex type theorems appearing in the literature, coming in both smooth and discrete flavors. The most familiar of these is the classical theorem in differential geometry, which states that the curvature function of a…
Newton's quadrilateral theorem can be phrased as follows. If H is a circle that is tangent to the four extended sides of a non-parallelogram quadrilateral Q, the center of H lies on the Newton line of Q. We prove that the theorem remains…
Four-Color Theorem has secret in its logical proof and actual operating. In this paper we will give a proof of Four-Color Theorem based on Kuratowski's Theorem using some induction argument and give a description of the most complicated…