L2WPMA / LEONARDO DA VINCI

NATIONAL TECHNICAL UNIVERSITY OF ATHENS SCHOOL OF ELECTRICAL AND COMPUTER ENGINEERING

ReENGINEERING THE ART THROUGH THE LENS OF L2WPMA:

A TRIBUTE TO LEONARDO DA VINCI’s INVENTIONS Flying Machines, War Machines, Architect/Innovations and Water & Land Machines

vol. 2 Dr. ATHANASIOS G. LAZAROPOULOS

Dr. Athanasios G. Lazaropoulos email: AGLazaropoulos@gmail.com Nov. 2014 The cover painting is based on the Leonardo da Vlncl’s painting entitled “Last Supper”

Introduction Piecewise Monotonic data Approximation (PMA) has been introduced by Demetriou and Powell in [1]. Among the various methods proposed to perform piecewise monotonic data approximations, Demejrlot’s L1PMA and L2WPMA immediately attract the attention of the academia [2], [3]. Actually, there is a great range of potential applications of the former two methods in signal restoration, image processing and spectroscopy. Recently, the artistic contribution of L1PMA has been highlighted in [4]. Now, in this art book, L2WPMA makes its debut in the artistic fields.

Similarly to L1PMA, L2WPMA solves data approximation problems using a number of monotonic sections predefined by the user.

Similarly to [4], various artistic techniques can also be adopted and combined with L2WPMA. This art album is dedicated to Leonardo da Vinci; an inventor ahead of his time.

Apart from his famots palnjlngs stch as “Lasj Stpper” of jhe couer, da Vlncl ls jhe “fajher” of many flying machines, war machines, architect/innovations and water & land machines. With reference to [5]-[7], some of da Vlncl’s mosj notable inventions are here highlighted through the lens of L2WPMA. In this arj book, jhe da Vlncl’s lngenlots charlsma ls comblned vljh jhe pure mathematics of L2WPMA providing an interesting engineering-based artistic view.

Contents 1. Flying Machines a. Flying Machine b. Helicopter c. Parachute d. Anemometer

2. War Machines 3. Architect/ a. 33-Barreled Organ Innovations b. Armored Car a. Clock c. Giant Crossbow b. Ideal City d. Triple Barrel c. Robotic Knight Cannon d. Self-Propelled Cart

4. Water & Land Machines a. Scuba Gear b. Revolving Bridge

Flying Machines Flying Machine Of da Vinci’s many areas of sjtdy, perhaps its favorite one was that of aviation.

One of da Vlncl’s mosj famots lnuenjlons, jhe flying machine (also known as the "ornithopter") ideally displays his powers of observation and imagination. ilso, da Vlncl’s flying machine reveals his enthusiasm for the potential of flight.

Manuscripts

Number of monotonic sections: 10 Approximation polarization: Mixed Contrast: No Real/Approximation residual: No Cube dimension: 1 pixel

Number of monotonic sections: 2 Approximation polarization: Mixed Contrast: No Real/Approximation residual: No Cube dimension: 1 pixel

Practical Implementation

Number of monotonic sections: 2, 3 and 4 Approximation polarization: Vertical Contrast: No Real/Approximation residual: No Cube dimension: 1 pixel

Flying Machines Helicopter Also known as the "Helical Air Screw", da Vlncl’s device was designed to compress air in order to obtain flight.

Da Vinci was a big proponent of the many possibilities offered by the screw shape since he uses it across many inventions and designs.

Manuscript

Number of monotonic sections: 10 Approximation polarization: Mixed Contrast: No Real/Approximation residual: No Cube dimension: 1 pixel

Practical Implementation

Number of monotonic sections: 3 and 4 Approximation polarization: Vertical Contrast: No Real/Approximation residual: No Cube dimension: 3 pixels

Number of monotonic sections: 5 and 6 Approximation polarization: Vertical Contrast: No Real/Approximation residual: No Cube dimension: 3 pixels

Flying Machines Parachute Da Vinci made a sketch of the first practical parachute with this accompanying comment: "If a man has a tent made of linen of which the apertures (openings) have all been stopped up, and it be twelve braccia (about 23 feet) across and twelve in depth, he will be able to throw himself down from any great height without suffering any injury."

Manuscript and Practical Implementation

Number of monotonic sections: 10 Approximation polarization: Mixed Contrast: No Real/Approximation residual: No Cube dimension: 3 pixels

Flying Machines Anemometer Leonardo da Vlncl’s fasclnajlon with flight inspired him to innovate an instrument for measuring the speed of wind.

Number of monotonic sections: 6 Approximation polarization: Mixed Real/Approximation residual: No Cube dimension: 1 pixel

Practical Implementation

Number of monotonic sections: 3 and 4 Approximation polarization: Horizontal Contrast: No Real/Approximation residual: No Cube dimension: 1 pixel

War Machines 33-Barreled Organ To solve the delay of canon loading, da Vinci proposed to build multi-barreled guns that could be loaded and fire simultaneously.

This idea forms the basis of war inventions llke da Vlncl’s 33-barreled organ, which featured 33 small-caliber guns connected together.

Manuscript

Number of monotonic sections: 10 Approximation polarization: Mixed Contrast: No Real/Approximation residual: No Cube dimension: 1 pixel

Practical Implementation

Number of monotonic sections: 3 Approximation polarization: Mixed Contrast: No Real/Approximation residual: No Cube dimension: 10 pixels

Practical Implementation

Number of monotonic sections: 3 Approximation polarization: Vertical Contrast: No Real/Approximation residual: Yes Cube dimension: 3 and 5 pixels

War Machines Armored Car Designed to intimidate an opposing army, the armored car is the most famous of da Vlncl’s war machines. Being precursor to the modern tank, the armored car invention was capable of moving in any direction and was equipped with a large number of weapons.

Manuscripts

Number of monotonic sections: 10 Approximation polarization: Mixed Contrast: No Real/Approximation residual: No Cube dimension: 1 pixel

Practical Implementation

Number of monotonic sections: 2 and 4 Approximation polarization: Vertical Contrast: No Real/Approximation residual: No Cube dimension: 2 pixels

Practical Implementation

Number of monotonic sections: 5 Approximation polarization: Horizontal Contrast: No Real/Approximation residual: No Cube dimension: 2 and 4 pixels

War Machines Giant Crossbow Designed for pure lnjlmldajlon, da Vlncl’s crossbow was to measure 42 braccia (or 27 yards) across.

Rather than fire giant arrows, da Vinci’s crossbov was designed to fire large stones or possibly flaming bombs.

Manuscript and Practical Implementation

Number of monotonic sections: 10 Approximation polarization: Mixed Contrast: No Real/Approximation residual: No Cube dimension: 4 pixels

War Machines Triple Barrel Cannon As a military engineer, one of da Vlncl’s key beliefs was that mobility was crucial to victory on the battlefield. Da Vinci designed his triple barrel canon to solve both the problems of fast reload and heavy weight.

Manuscript and Practical Implementation

Number of monotonic sections: 4 Approximation polarization: Mixed Contrast: No Real/Approximation residual: No Cube dimension: 1 pixel

Architect/Innovations Clock Da Vinci did not invent the clock but what he did was to design a more accurate one. Da Vinci’s clock had jvo separaje mechanlsms: one for minutes and one for hours.

Manuscript and Practical Implementation

Number of monotonic sections: 3 Approximation polarization: Vertical Contrast: No Real/Approximation residual: No Cube dimension: 3 pixels

Architect/Innovations Ideal City Perhaps da Vlncl’s lnuenjlon of ideal city better speaks its epic ambitions and life scope. This invention focuses not just on a single area but on a wider one combining all of da Vlncl’s jalenjs as an artist, architect, engineer and inventor.

Manuscripts

Number of monotonic sections: 10 Approximation polarization: Mixed Contrast: No Real/Approximation residual: No Cube dimension: 1 pixel

Manuscripts and Practical Implementation

Number of monotonic sections: 10 Approximation polarization: Vertical Contrast: No Real/Approximation residual: No Cube dimension: 2 pixels

Architect/Innovations Robotic Knight Da Vinci had many ideas that employed the use of pulleys, weights and gears. Da Vinci incorporated these three components into his Robotic Knight which many people consider the first robot.

Thotgh a ftll dravlng of da Vlncl’s robojlc knlghj has never been recovered, fragments detailing different aspects of the knight have been discovered throughout his manuscripts.

Manuscripts

Number of monotonic sections: 10 Approximation polarization: Mixed Contrast: No Real/Approximation residual: No Cube dimension: 3 pixels

Architect/Innovations Self-Propelled Cart Many consider da Vinci’s self-propelled cart as jhe vorld’s first motorized vehicle. Da Vinci designed a self-propelled cart capable of moving without being pushed.

Manuscript and Practical Implementation

Number of monotonic sections: 2 Approximation polarization: Vertical Contrast: Invert Colors Real/Approximation residual: Yes Cube dimension: 2 pixels

Water & Land Machines Scuba Gear While working in Venice, the "water city", da Vinci designed his scuba gear. His intention was to create the proper underwater equipment for sneak attacks on enemy ships.

The leather diving suit was equipped with a bag-llke mask jhaj venj ouer jhe dluer’s head.

Two cane tubes were attached to the mask around the nose that led up to a surface box.

Manuscript and Practical Implementation

Number of monotonic sections: 8 Approximation polarization: Mixed Contrast: No Real/Approximation residual: No Cube dimension: 3 pixels

Water & Land Machines Revolving Bridge Deslgned for Dtke Sforka, da Vlncl’s revolving bridge could be quickly packed up and transported. His revolving bridge could be used by armies on the move to pass over rivers and lakes.

Manuscript and Practical Implementation

Number of monotonic sections: 10 Approximation polarization: Mixed Contrast: No Real/Approximation residual: No Cube dimension: 3 pixels

Practical Implementation

Number of monotonic sections: 6 Approximation polarization: Mixed Contrast: No Real/Approximation residual: No Cube dimension: 2 pixels

References [1] I. C. Demejrlot and M. J. D. Oovell, “Leasj sqtares smoojhlng of tnluarlaje daja jo achleue plecevlse monojonlcljy”, IMA J. of Numerical Analysis, vol. 11, pp. 411-432, 1991.

[2] I. C. Demejrlot, “L1OMi: i Forjran 77 Oackage for Besj L1 Olecevlse Monojonlc Daja Smoojhlng”, Computer Physics Communications, vol. 151, no. 1, pp. 315-338, 2003.

[3] I. C. Demejrlot, “Algorithm 863: L2WPMA, a Fortran 77 package for weighted least-squares piecewise monotonic data approximation." ACM Transactions on Mathematical Software (TOMS), vol. 33, no. 1, pp 1-19, Mar. 2007.

[4] http://issuu.com/lazaropoulos/docs/l1pma [5] http://www.da-vinci-inventions.com/ [6] http://en.wikipedia.org/wiki/Science_and_inventions_of_Leonardo_da_Vinci [7] http://www.leonardodavincisinventions.com/