Monday, February 1, 2021

The 'Cube' in Albrecht Durer's Engraving, 'Melancholia', 1514

This is the kind of research that happens when one is self-isolating through a winter pandemic.   

Albrecht Durer's engraving 'Melancholia', was first printed in 1514. 

 I wrote about the carpentry tools scattered around the edges of the image in my previous post: "Albrecht Durer's 'Melancholia' and His Knowledge of Construction and Practical Geometry", posted on 1/17/2021.

On the left side in the middle is a polyhedron, the 'Cube'^, a 3-d shape.The angel is studying it intently. Why? And what is it?

And why the sphere? A pure geometric shape, white, abstract, perfect, set among real tools we can identify, and a skinny dog. It is the only other abstract thing besides the Cube in the engraving. 

I have some suggestions based on Durer's knowledge of and interest in geometry.


Durer wrote about geometry in his book, Underweysung der Mesang mit dem Zirckel und Richtscheyt,  published in 1525.*   The last 6 words of the title translate as Measurement with Compass and Straightedge


 One of his diagrams:


I copied the pattern of triangles, cut it out, and folded it into a  20 sided polyhedron, (an icosahedron,) fastened it with tape. 

Note: The 5 triangles on the right that would complete the shape are not taped together; it fits in my hand.

If the shape were made from wire it would look like the drawings beside the pattern - the view from the sides and then from the top or the bottom. The lines are the edges of the polygons. 



Another of Durer's diagrams is 12 joined pentagons which become a polyhedron. To the right are the side and top views. The views are not elevations; they are transparent.  


Again, I copied it and cut it out.




It folds up to be much like a soccer ball with edges.

Durer's book contains many of these polyhedrons. He mixes the kinds of polygons to make the 'sphere'. Here are 2 of his simple ones



The models led me to consider the Cube in Melancholia as a study about how to create a sphere from planes, from polygons. 


I extended the main sides of the polygon - red lines - so that the shapes became diamonds. Then I extended the edge of the almost invisible left side and found its tip met the main side at the top. 






Next I divided the length of 2 sides of the diamonds in half. When I joined the mid points with a red lines  I saw that they followed the edge of Durer's polyhedron.




Perhaps Durer was drawing a truncated cube in perspective.  I made a diagram: 6 squares which if folded would make a cube. I lopped off 6 corners  - red dashed lines




I added the triangles, printed it, and cut it out.





Cut and fastened with tape, it is flimsy, not  solid like Durer's block.





Tipped on its side, it has the planes of Durer's 'Cube' - his 8 sided shape.











Historians tend to title the book, Measurement with Compass and Ruler. In 1525, rods with 10 parts were common, so were boards with straight edges, used for drawing straight lines. Regular, agreed upon dimensions, such as would be on a ruler, did not exist. Note that there is no ruler among the tools Durer includes in his engraving, 'Melancholia'. 

^ Durer's polyhedron is not a cube. I use the name as an easy reference.

*Underweysung der Mesang mit dem Zirckel und Richtscheyt can be read online through the Warnock Library in Nuremberg, Germany.  A translation by Walter Strauss, published in 1977, is out of print, only available at museum and college libraries which currently are closed.

At any time, but especially in the year of Covid-19, to read a 500 year old book, housed 3800 miles away, while sitting in my office with my cat sprawled out beside me, is remarkable. That I can shift from one page to another and back again as I consider the images is an amazing gift.

Sunday, January 17, 2021

Albrecht Durer's 'Melancholia' and his knowledge of construction and practical geometry

Note: click on any image to enlarge it!

Albrecht Durer was a painter and print maker in Germany, 1471-1528. He also wrote books, and traveled widely in western Europe. He was a superb draftsman. I have enjoyed the compositions, the details, and the lines in his engravings and woodcuts for 50 years.

His woodcuts were for the people, most of whom were  illiterate. Their livelihoods did not require writing. They could read the images: here the rough stable, the well dressed men coming to see a baby, the angels and the star. 

The engravings were not so easy to make. They were for books, for people who could read.

These plates come from the Dover Publications reprints of Durer's wood cuts and engravings.* This is print #183.



He drew what he saw around him. His plates are full of the life he knew, including construction details. 

This detail from #183 shows a truss which includes a collar tie with angled tenon joints and pegs. The purlins and rafters are structurally correct.



The detail from Plate #190, shows the grain of the wood brace running in the right direction. The angled cuts for the joints could serve as templates for repair.


    Here in Plate #185, the brace is tied. a peg serves to tighten the tie as needed. The thatch for the roof is properly applied.

    When I began to write this post I was thinking about Durer's knowledge of construction and his use of geometry in his compositions. (More about that in another post.)
    I was side tracked as I began to read Durer biographies. The scholars who wrote them rarely knew about construction. To them the structures are allegories or useful for his compositions. 
    I saw practical and abstract geometry.
    This is my exploration, as a Geometer, of one of  Durer's most important engravings.

    'Melancholia', Durer's engraving about the temperament of  artists and  artisans, was made in 1514.*  

    Durer put many construction tools around his melancholy angel. She holds a compass.  Tucked beside her hand is a gauge. The putto sits on a mill wheel next to a ladder. Set beside the polyhedron is a pot for hot liquid metal, possibly lead, on a brazier. Under the angel's skirts is a pair of pliers.

    The tools are used in practical geometry. Above the dog is a hammer. Between his hind legs and the sphere: a Line and its plumb bob.

    In the lower left corner is a profile gauge.

    Then: a plane, a saw, and a straight edge that can be used to draw arcs. It may also be a level.

    Lastly: nails and a nail punch.

    The theory was that Melancholy, influenced by the planet Saturn, was part of the inherent character of artistic and philosophical people. There were 3 levels. The lowest was artists and artisans. The next was scholars, natural scientists, and statesmen. The highest level was theologians and those who studied the secrets of the divine.
    Here the putto is taking a nap after doing some numbers (perhaps) on a slate, the lowest level. The angel, on the other hand, is intently studying the polyhedron, an abstract shape. She is a scientist. The dog  - which I learned represents 'faithfulness' - is waiting. The sphere is an abstraction, perfect as the tools and mill wheel are not. They are all 'things', 


    The scales, the hour glass, the bell are said to refer to the knowledge of artisans: they understand weight, measure (of time), music (as an expression of geometry). 


    I have very little understanding of the Magic Square. I do know that all the lines add up to 34 and no number is used twice.

    I do wonder if in a largely illiterate society, the fairly recent acceptance of Hindu-Arabic numbers, as opposed to Roman Numerals, might be part of its purpose here: to illustrate the scholarship, the mathematics that numbers made possible.

    Try substituting Roman Numerals in the grid of the Square.
    Can you quickly add up the amounts? How would you do it as an arithmetic problem on paper? 

     16 +5+9+4 = XVI+V+IX+IV .


    I will write about the polyhedron in the next post. 


    *images from  The Complete Woodcuts of Albretch Durer, edited by Dr. Willi Kurth, 1963  and The Complete Engravings, Etchings & Drypoints of Albrecht Durer, edited by Walter L Strauss, 1972.  Both republished by Dover Publications: www.dovderpublications. com.

    Walter L. Strauss' analysis is the best I've found. I wish he had known more about geometry. 

    Friday, November 13, 2020

    ARCHITECTURAL GEOMETRY, A Rare Geometrical Record from Rural Devon, by Laurie Smith



    Laurie Smith has written a new book: ARCHITECTURAL  GEOMETRY  A Rare Geometrical Record from Rural Devon.  

    Here's the cover.











    The book is about the many daisy wheels and other geometry found on the walls of a Devon threshing barn.
    The barn, shown here, is owned by Richard Westcott, editor of The Three Hares, a Curiosity Worth Regarding. 
    The image is #3, page 4.  

    Richard Westcott, Laurie Smith, and their friend, the photographer and film maker Chris Chapman, examined and recorded the geometry on the barn’s walls -  over 169 separate geometric shapes.

    This photograph is of a "divider scribed daisy wheel from the wall's inner surface." 

    The image is #4; the quote is from page 4.

    They researched the barn’s history, took measured drawings and photographs, and explored the geometry.

    Then Laurie wrote this book. 

    Like all of Laurie’s books it has beautiful diagrams. Clear descriptions accompanying the diagrams explain how the daisy wheels still visible on its walls governed the siting, layout, and frame  of the barn.

    Image 39, page 26

    This is 1of 5 of Laurie's illustrations showing the development of the barn section.     

    He includes examples of similar geometries give context and nuance. 

    Here is one of 4 daisy wheel drawings for the geometry of the Barley Barn, Cressing Temple, Essex, UK

     Image 27, page 19.





    Along the way Laurie explains terms and forms which we rarely use today, including the use of a perch, pole and rod as measuring devices.  

    He introduces the reader to the Trivium, the Quadrivium, and Whirling Squares. 

    Part of Image 63, page 50 


    He writes thoughtfully - and with humor -  about apotropaic symbols.  



    At the end of the book Laurie considers how all of that geometry - 169 separate images - came to be scribed on the interior walls of a rural threshing barn. He suggests a 'geometry school'.  I agree with his theory: I have also found incomplete geometries drawn on plaster walls here in the States.

    His descriptions encourage the reader to examine the image, think about what he's written, look again, and understand the geometry.

    Here is what he says about this double daisy wheel:

    "The image shows the geometrical precision of the divider-scribing, the scars of the divider pin at the twelve points around the primary circle and the compound damage caused at the symbol's axis by the passing of twenty four arcs."

    Image and quote, page 59 


    The book's copyright page includes this introduction: 

    "Laurie Smith is an independent early-building design researcher, specializing in geometric design systems. Because the medieval educational curriculum included geometry he uses geometric analysis to excavate and recover the design systems of the past, a process he thinks of as design archaeology. He lectures, writes, runs practical workshops  and publishes educational articles on geometrical design that are available from his website."  


     I highly recommend this book to all who are interested in historic construction and geometry. Copies can be purchased from Laurie Smith in the UK or from me ($20.00 postage paid) in the USA. 


    *All the photographs: the barn, the hand and the daisy wheel, and the 12 pointed daisy wheel in the barn shown here are by Chris Chapman, copied by me from the book for the purpose of this review. 

    The geometry is by Laurie Smith, also copied with his permission.