​Kim Williams is a writer and editor living and working in Italy. She received her degree in Architectural Studies from the University of Texas in Austin, and is licensed as an architect in New York State. Her apprenticeship was done in the offices of Philip Johnson in New York. She became interested in mathematics and architecture while writing Italian Pavements: Patterns in Space (Houston: Anchorage Press, 1997) about the role of decorated pavements in the history of Italian architecture. In 1996 she began the international conference series “Nexus: Architecture and Mathematics”, the thirteenth edition of which, Nexus 2020/21, will take place in Kaiserlautern, Germany, in July 2021. In 1999 she founded the Nexus Network Journal to provide a dedicated venue for scholarly research in architecture and mathematics.In 2000 she founded Kim Williams Books, an independent press for books about architecture and mathematics. Kim has published many articles in scholarly journals on the use of mathematical principles in architecture, including Mathematical Intelligencer and Leonardo. Her drawings have been displayed in both group and solo exhibits. She has participated in numerous international conferences. She co-edited, with Michael Ostwald, the 2-volume Architecture and Mathematics from Antiquity to the Future (Birkhäuser, 2015). She is editor for Springer of the book series "Mathematics and the Built Environment". Her latest book is Daniele Barbaro’s Vitruvius of 1567.Cosimo Monteleone is Associate Professor of Descriptive Geometry and Architectural Representation at the Università degli Studi di Padova. In 2003 he obtained his degree in Architecture at the University IUAV of Venice, where he also earned his Ph.D. in 2010 in “Architectural Composition curriculum in Survey and Representation of Architecture and Landscape” with a study on the Guggenheim Museum by Frank Lloyd Wright. His research focuses on architectural, urban and landscape survey; 3D modelling of architecture and city; augmented and virtual reality; gnomonic; science and technique applied to art and architecture; history of representation with particular skills related to Renaissance perspective. He is member of “Visualizing Cities”, an international research devoted to the analysis and the representation of the historical, urban and architectural transformations. He is member of the National Technical UNI - UNI / CT 047 / GL 03 (Technical drawing for building and installations). On the topics of his research he has published several essays, presented conferences, lectures, directed digital installations for national and international exhibitions. Among his publications is Frank Lloyd Wright. Geometria e Astrazione nel Guggenheim Museum (Rome: Aracne 2013). He has published two papers in the Nexus Network Journal: “Perspective at Palladio’s Time and Its Scientific Heritage” (with Andrea Giordano) and “The Perspective of Daniele Barbaro” (2019). His latest book is La prospettiva di Daniele Barbaro. Note critiche e trascrizione del manoscritto It. IV, 39=5446 (Aracne, 2020).

• Preface.- Translators’ Note.- Acknowledgements.- Daniele Barbaro and the Geometric Solids.- Daniele Barbaro’s Innovations in Perspective Studies.- Dedication to Matteo Macigni.- Foreword.- Part I, setting up a perspective.- The Ordering of the principles.- On the eye.- On the way of seeing.- On the thing seen.- On distance.- On the division of planes.- Where the eye must be placed.- On the distance.- Of the size that figures must be made in the painting.- Part II, in which are treated ichnographia, that is, the description of the plan.- The practice of describing figures.- The way of describing plans.- The way to degrade a given plane.- The way to reduce the degraded plane into a square.- Division of the degraded square according to the perfect square.- The way to add to or take away from the degraded square.- How to cut a square from a quadrangular surface that is wider than it is long.- How to respond to those who, in dividing the plane into braccia, cometo have a foreshortening that is larger than the perfect.- The plan of the triangle, and how the degraded is made from the perfect form.- How to form the plan of a cube.- Other ways of making plans.- Part III, which treats the ways of raising the body from the plan.- Three ways of raising bodies from the plans.- Unfolding, raising and shadowing the pyramid.- Unfolding, raising and shadowing the cube.- Unfolding, raising and shadowing the body called ‘octahedron’.- Unfolding, raising and shadowing the dodecahedral body.- Unfolding, raising and shadowing the body called ‘icosahedron’.- Description of the irregular bodies that are born from the regular bodies [truncated tetrahedron].- Description of a body that is born from the cube and from the octahedron, and its unfolding [cubocatahedron].- Description of another irregular body that is born from the cube [truncated cube].- Description and unfolding of another body thatis born from the cube [small rhombicuboctahedron].- Unfolding and description of a body that is born from the octahedron [truncated octahedron].- Unfolding and description of a body that is born from the dodecahedron and from the icosahedron [icosidodecahedron].- Unfolding and description of another body that is born from the dodecahedron [truncated dodecahedron].- Unfolding and description of another body that is born from the icosahedron [truncated icosahedron].- Unfolding and description of another body that is born from a composite body [great rhombicuboctahedron].- Unfolding and description of another body that is born from a composite body  [rectified truncated octahedron].- Unfolding and description of another body composite body [small rhombicosidodecahedron].- Unfolding and description of another body [rectified truncated icosahedron].- Unfolding of some other bodies [truncated pentakis dodecahedron].- Unfolding of another irregular body [truncated truncated icosahedron].- Unfolding of some other bodies [great rhombicosidodecahedron].- Unfolding of another irregular body [truncated truncated octahedron].- Unfolding of another irregular body [great rhombicosidodecahedron augmented with pentagonal cupolas].- Unfolding of another irregular body [great rhombicuboctahedron augmented with square cupolas].- Unfolding of another irregular body.- Unfolding of some bodies based on the surfaces of bodies both regular and irregular, and first that which is based on the pyramid [elevated tetrahedron].- Unfolding of a body based on the cube [elevated cube].- Unfolding of a body based on the octagon [elevated octahedron].- Unfolding of a body based on the dodecahedron [elevated dodecahedron].- Unfolding of a body based on the icosahedron [elevated icosahedron].- Unfolding of a body based on an irregular body [elevated cuboctahedron].-Unfolding of a body resting on an underlying body [elevated small rhombicuboctahedron].- Unfolding of another body [elevated icosidodecahedron].- Description of some of the previous bodies with pyramids.- Some easy and quick ways to draw plans and bodies.- How to degrade many circles drawn on the same centre and divided into several parts.- An easy way to degrade without rulers.- Description of the torchio, or mazzocchio.- The perfect description of the mazzocchio.- Part Four, in which will be treated scenographia, that is, the description of scenes.- The manners of scenes.- Description and shading of the Tuscan base.-  Description and shading of the Attic base.- Description and description of the Ionic base.- Description and degradation of columns.- Description and degradation of the Tuscan capital.-  Description and degradation of the Doric capital.-Description and degradation of the Ionic capital.- Description and degradation of the Corinthian capital.- Description of the Doric architrave.- Description of the Ionic architrave and the pediment.- Rules for the heights of the architraves and ways to transpose them from one size to another.- The description and degradation of stairs.- Of arches, vaults and doors.- Plan, elevation and profile of a temple.- Description of the Tragic scene.- Description of the Comic scene.- Description of the Satiric scene.- Part V, in which is set forth a fine and secret part of perspective.- [Distorting images] .- Part VI, which is called Planispherio.- Unfolding, description and degradation of the sphere.- Projection of the sphere onto the plane according to the ancients.- How to make projections of oblique circles.- The projection of the circles of the heights onto the plane.- Part VII, which treats lighting, shadows and colours.- [The importance of light, colour and chiaroscuro].- On the projection of shadows.- Part VIII, in which are treated the measures of the human body.- Measurement of the human body.- The way to describe the plan of the human head.- How to describe heads that are neither in profile nor straight on.- The way to make heads that look either up or down.- Part IX, in which are described many instruments and ways of rendering things in perspective and transporting them.- The author’s instrument for making clocks in every climate, on every plane and for every sort of hour.- The use of the universal clock.- Albrecht Dürer’s instrument for making perspectives.- Building another instrument by Baldassarre Lanci.- Natural manners of rendering things in perspective.- [Appendix] How with a new instrument it is possible to know the inclination of scarps of walls, according to the invention of Giacomo Castriotto.- Index.

​Kim Williams is a writer and editor living and working in Italy. She received her degree in Architectural Studies from the University of Texas in Austin, and is licensed as an architect in New York State. Her apprenticeship was done in the offices of Philip Johnson in New York. She became interested in mathematics and architecture while writing Italian Pavements: Patterns in Space (Houston: Anchorage Press, 1997) about the role of decorated pavements in the history of Italian architecture. In 1996 she began the international conference series “Nexus: Architecture and Mathematics”, the thirteenth edition of which, Nexus 2020/21, will take place in Kaiserlautern, Germany, in July 2021. In 1999 she founded the Nexus Network Journal to provide a dedicated venue for scholarly research in architecture and mathematics.In 2000 she founded Kim Williams Books, an independent press for books about architecture and mathematics. Kim has published many articles in scholarly journals on the use of mathematical principles in architecture, including Mathematical Intelligencer and Leonardo. Her drawings have been displayed in both group and solo exhibits. She has participated in numerous international conferences. She co-edited, with Michael Ostwald, the 2-volume Architecture and Mathematics from Antiquity to the Future (Birkhäuser, 2015). She is editor for Springer of the book series "Mathematics and the Built Environment". Her latest book is Daniele Barbaro’s Vitruvius of 1567.Cosimo Monteleone is Associate Professor of Descriptive Geometry and Architectural Representation at the Università degli Studi di Padova. In 2003 he obtained his degree in Architecture at the University IUAV of Venice, where he also earned his Ph.D. in 2010 in “Architectural Composition curriculum in Survey and Representation of Architecture and Landscape” with a study on the Guggenheim Museum by Frank Lloyd Wright. His research focuses on architectural, urban and landscape survey; 3D modelling of architecture and city; augmented and virtual reality; gnomonic; science and technique applied to art and architecture; history of representation with particular skills related to Renaissance perspective. He is member of “Visualizing Cities”, an international research devoted to the analysis and the representation of the historical, urban and architectural transformations. He is member of the National Technical UNI - UNI / CT 047 / GL 03 (Technical drawing for building and installations). On the topics of his research he has published several essays, presented conferences, lectures, directed digital installations for national and international exhibitions. Among his publications is Frank Lloyd Wright. Geometria e Astrazione nel Guggenheim Museum (Rome: Aracne 2013). He has published two papers in the Nexus Network Journal: “Perspective at Palladio’s Time and Its Scientific Heritage” (with Andrea Giordano) and “The Perspective of Daniele Barbaro” (2019). His latest book is La prospettiva di Daniele Barbaro. Note critiche e trascrizione del manoscritto It. IV, 39=5446 (Aracne, 2020).

• Preface.- Translators’ Note.- Acknowledgements.- Daniele Barbaro and the Geometric Solids.- Daniele Barbaro’s Innovations in Perspective Studies.- Dedication to Matteo Macigni.- Foreword.- Part I, setting up a perspective.- The Ordering of the principles.- On the eye.- On the way of seeing.- On the thing seen.- On distance.- On the division of planes.- Where the eye must be placed.- On the distance.- Of the size that figures must be made in the painting.- Part II, in which are treated ichnographia, that is, the description of the plan.- The practice of describing figures.- The way of describing plans.- The way to degrade a given plane.- The way to reduce the degraded plane into a square.- Division of the degraded square according to the perfect square.- The way to add to or take away from the degraded square.- How to cut a square from a quadrangular surface that is wider than it is long.- How to respond to those who, in dividing the plane into braccia, cometo have a foreshortening that is larger than the perfect.- The plan of the triangle, and how the degraded is made from the perfect form.- How to form the plan of a cube.- Other ways of making plans.- Part III, which treats the ways of raising the body from the plan.- Three ways of raising bodies from the plans.- Unfolding, raising and shadowing the pyramid.- Unfolding, raising and shadowing the cube.- Unfolding, raising and shadowing the body called ‘octahedron’.- Unfolding, raising and shadowing the dodecahedral body.- Unfolding, raising and shadowing the body called ‘icosahedron’.- Description of the irregular bodies that are born from the regular bodies [truncated tetrahedron].- Description of a body that is born from the cube and from the octahedron, and its unfolding [cubocatahedron].- Description of another irregular body that is born from the cube [truncated cube].- Description and unfolding of another body thatis born from the cube [small rhombicuboctahedron].- Unfolding and description of a body that is born from the octahedron [truncated octahedron].- Unfolding and description of a body that is born from the dodecahedron and from the icosahedron [icosidodecahedron].- Unfolding and description of another body that is born from the dodecahedron [truncated dodecahedron].- Unfolding and description of another body that is born from the icosahedron [truncated icosahedron].- Unfolding and description of another body that is born from a composite body [great rhombicuboctahedron].- Unfolding and description of another body that is born from a composite body  [rectified truncated octahedron].- Unfolding and description of another body composite body [small rhombicosidodecahedron].- Unfolding and description of another body [rectified truncated icosahedron].- Unfolding of some other bodies [truncated pentakis dodecahedron].- Unfolding of another irregular body [truncated truncated icosahedron].- Unfolding of some other bodies [great rhombicosidodecahedron].- Unfolding of another irregular body [truncated truncated octahedron].- Unfolding of another irregular body [great rhombicosidodecahedron augmented with pentagonal cupolas].- Unfolding of another irregular body [great rhombicuboctahedron augmented with square cupolas].- Unfolding of another irregular body.- Unfolding of some bodies based on the surfaces of bodies both regular and irregular, and first that which is based on the pyramid [elevated tetrahedron].- Unfolding of a body based on the cube [elevated cube].- Unfolding of a body based on the octagon [elevated octahedron].- Unfolding of a body based on the dodecahedron [elevated dodecahedron].- Unfolding of a body based on the icosahedron [elevated icosahedron].- Unfolding of a body based on an irregular body [elevated cuboctahedron].-Unfolding of a body resting on an underlying body [elevated small rhombicuboctahedron].- Unfolding of another body [elevated icosidodecahedron].- Description of some of the previous bodies with pyramids.- Some easy and quick ways to draw plans and bodies.- How to degrade many circles drawn on the same centre and divided into several parts.- An easy way to degrade without rulers.- Description of the torchio, or mazzocchio.- The perfect description of the mazzocchio.- Part Four, in which will be treated scenographia, that is, the description of scenes.- The manners of scenes.- Description and shading of the Tuscan base.-  Description and shading of the Attic base.- Description and description of the Ionic base.- Description and degradation of columns.- Description and degradation of the Tuscan capital.-  Description and degradation of the Doric capital.-Description and degradation of the Ionic capital.- Description and degradation of the Corinthian capital.- Description of the Doric architrave.- Description of the Ionic architrave and the pediment.- Rules for the heights of the architraves and ways to transpose them from one size to another.- The description and degradation of stairs.- Of arches, vaults and doors.- Plan, elevation and profile of a temple.- Description of the Tragic scene.- Description of the Comic scene.- Description of the Satiric scene.- Part V, in which is set forth a fine and secret part of perspective.- [Distorting images] .- Part VI, which is called Planispherio.- Unfolding, description and degradation of the sphere.- Projection of the sphere onto the plane according to the ancients.- How to make projections of oblique circles.- The projection of the circles of the heights onto the plane.- Part VII, which treats lighting, shadows and colours.- [The importance of light, colour and chiaroscuro].- On the projection of shadows.- Part VIII, in which are treated the measures of the human body.- Measurement of the human body.- The way to describe the plan of the human head.- How to describe heads that are neither in profile nor straight on.- The way to make heads that look either up or down.- Part IX, in which are described many instruments and ways of rendering things in perspective and transporting them.- The author’s instrument for making clocks in every climate, on every plane and for every sort of hour.- The use of the universal clock.- Albrecht Dürer’s instrument for making perspectives.- Building another instrument by Baldassarre Lanci.- Natural manners of rendering things in perspective.- [Appendix] How with a new instrument it is possible to know the inclination of scarps of walls, according to the invention of Giacomo Castriotto.- Index.

​Kim Williams is a writer and editor living and working in Italy. She received her degree in Architectural Studies from the University of Texas in Austin, and is licensed as an architect in New York State. Her apprenticeship was done in the offices of Philip Johnson in New York. She became interested in mathematics and architecture while writing Italian Pavements: Patterns in Space (Houston: Anchorage Press, 1997) about the role of decorated pavements in the history of Italian architecture. In 1996 she began the international conference series “Nexus: Architecture and Mathematics”, the thirteenth edition of which, Nexus 2020/21, will take place in Kaiserlautern, Germany, in July 2021. In 1999 she founded the Nexus Network Journal to provide a dedicated venue for scholarly research in architecture and mathematics.In 2000 she founded Kim Williams Books, an independent press for books about architecture and mathematics. Kim has published many articles in scholarly journals on the use of mathematical principles in architecture, including Mathematical Intelligencer and Leonardo. Her drawings have been displayed in both group and solo exhibits. She has participated in numerous international conferences. She co-edited, with Michael Ostwald, the 2-volume Architecture and Mathematics from Antiquity to the Future (Birkhäuser, 2015). She is editor for Springer of the book series "Mathematics and the Built Environment". Her latest book is Daniele Barbaro’s Vitruvius of 1567.Cosimo Monteleone is Associate Professor of Descriptive Geometry and Architectural Representation at the Università degli Studi di Padova. In 2003 he obtained his degree in Architecture at the University IUAV of Venice, where he also earned his Ph.D. in 2010 in “Architectural Composition curriculum in Survey and Representation of Architecture and Landscape” with a study on the Guggenheim Museum by Frank Lloyd Wright. His research focuses on architectural, urban and landscape survey; 3D modelling of architecture and city; augmented and virtual reality; gnomonic; science and technique applied to art and architecture; history of representation with particular skills related to Renaissance perspective. He is member of “Visualizing Cities”, an international research devoted to the analysis and the representation of the historical, urban and architectural transformations. He is member of the National Technical UNI - UNI / CT 047 / GL 03 (Technical drawing for building and installations). On the topics of his research he has published several essays, presented conferences, lectures, directed digital installations for national and international exhibitions. Among his publications is Frank Lloyd Wright. Geometria e Astrazione nel Guggenheim Museum (Rome: Aracne 2013). He has published two papers in the Nexus Network Journal: “Perspective at Palladio’s Time and Its Scientific Heritage” (with Andrea Giordano) and “The Perspective of Daniele Barbaro” (2019). His latest book is La prospettiva di Daniele Barbaro. Note critiche e trascrizione del manoscritto It. IV, 39=5446 (Aracne, 2020).

• Preface.- Translators’ Note.- Acknowledgements.- Daniele Barbaro and the Geometric Solids.- Daniele Barbaro’s Innovations in Perspective Studies.- Dedication to Matteo Macigni.- Foreword.- Part I, setting up a perspective.- The Ordering of the principles.- On the eye.- On the way of seeing.- On the thing seen.- On distance.- On the division of planes.- Where the eye must be placed.- On the distance.- Of the size that figures must be made in the painting.- Part II, in which are treated ichnographia, that is, the description of the plan.- The practice of describing figures.- The way of describing plans.- The way to degrade a given plane.- The way to reduce the degraded plane into a square.- Division of the degraded square according to the perfect square.- The way to add to or take away from the degraded square.- How to cut a square from a quadrangular surface that is wider than it is long.- How to respond to those who, in dividing the plane into braccia, cometo have a foreshortening that is larger than the perfect.- The plan of the triangle, and how the degraded is made from the perfect form.- How to form the plan of a cube.- Other ways of making plans.- Part III, which treats the ways of raising the body from the plan.- Three ways of raising bodies from the plans.- Unfolding, raising and shadowing the pyramid.- Unfolding, raising and shadowing the cube.- Unfolding, raising and shadowing the body called ‘octahedron’.- Unfolding, raising and shadowing the dodecahedral body.- Unfolding, raising and shadowing the body called ‘icosahedron’.- Description of the irregular bodies that are born from the regular bodies [truncated tetrahedron].- Description of a body that is born from the cube and from the octahedron, and its unfolding [cubocatahedron].- Description of another irregular body that is born from the cube [truncated cube].- Description and unfolding of another body thatis born from the cube [small rhombicuboctahedron].- Unfolding and description of a body that is born from the octahedron [truncated octahedron].- Unfolding and description of a body that is born from the dodecahedron and from the icosahedron [icosidodecahedron].- Unfolding and description of another body that is born from the dodecahedron [truncated dodecahedron].- Unfolding and description of another body that is born from the icosahedron [truncated icosahedron].- Unfolding and description of another body that is born from a composite body [great rhombicuboctahedron].- Unfolding and description of another body that is born from a composite body  [rectified truncated octahedron].- Unfolding and description of another body composite body [small rhombicosidodecahedron].- Unfolding and description of another body [rectified truncated icosahedron].- Unfolding of some other bodies [truncated pentakis dodecahedron].- Unfolding of another irregular body [truncated truncated icosahedron].- Unfolding of some other bodies [great rhombicosidodecahedron].- Unfolding of another irregular body [truncated truncated octahedron].- Unfolding of another irregular body [great rhombicosidodecahedron augmented with pentagonal cupolas].- Unfolding of another irregular body [great rhombicuboctahedron augmented with square cupolas].- Unfolding of another irregular body.- Unfolding of some bodies based on the surfaces of bodies both regular and irregular, and first that which is based on the pyramid [elevated tetrahedron].- Unfolding of a body based on the cube [elevated cube].- Unfolding of a body based on the octagon [elevated octahedron].- Unfolding of a body based on the dodecahedron [elevated dodecahedron].- Unfolding of a body based on the icosahedron [elevated icosahedron].- Unfolding of a body based on an irregular body [elevated cuboctahedron].-Unfolding of a body resting on an underlying body [elevated small rhombicuboctahedron].- Unfolding of another body [elevated icosidodecahedron].- Description of some of the previous bodies with pyramids.- Some easy and quick ways to draw plans and bodies.- How to degrade many circles drawn on the same centre and divided into several parts.- An easy way to degrade without rulers.- Description of the torchio, or mazzocchio.- The perfect description of the mazzocchio.- Part Four, in which will be treated scenographia, that is, the description of scenes.- The manners of scenes.- Description and shading of the Tuscan base.-  Description and shading of the Attic base.- Description and description of the Ionic base.- Description and degradation of columns.- Description and degradation of the Tuscan capital.-  Description and degradation of the Doric capital.-Description and degradation of the Ionic capital.- Description and degradation of the Corinthian capital.- Description of the Doric architrave.- Description of the Ionic architrave and the pediment.- Rules for the heights of the architraves and ways to transpose them from one size to another.- The description and degradation of stairs.- Of arches, vaults and doors.- Plan, elevation and profile of a temple.- Description of the Tragic scene.- Description of the Comic scene.- Description of the Satiric scene.- Part V, in which is set forth a fine and secret part of perspective.- [Distorting images] .- Part VI, which is called Planispherio.- Unfolding, description and degradation of the sphere.- Projection of the sphere onto the plane according to the ancients.- How to make projections of oblique circles.- The projection of the circles of the heights onto the plane.- Part VII, which treats lighting, shadows and colours.- [The importance of light, colour and chiaroscuro].- On the projection of shadows.- Part VIII, in which are treated the measures of the human body.- Measurement of the human body.- The way to describe the plan of the human head.- How to describe heads that are neither in profile nor straight on.- The way to make heads that look either up or down.- Part IX, in which are described many instruments and ways of rendering things in perspective and transporting them.- The author’s instrument for making clocks in every climate, on every plane and for every sort of hour.- The use of the universal clock.- Albrecht Dürer’s instrument for making perspectives.- Building another instrument by Baldassarre Lanci.- Natural manners of rendering things in perspective.- [Appendix] How with a new instrument it is possible to know the inclination of scarps of walls, according to the invention of Giacomo Castriotto.- Index.